GIFT  OF 


ROBINSON'S  MATHEMATICAL  SERIES. 


NEW 


UNIVERSITY   ALGEBRA: 


THEORETICAL  AND  PRACTICAL  TREATISE, 


MANY  NEW  AND   ORIGINAL  METHODS  AND  APPUCATl 


COLLEGES  AND  HIGH  SCHOOLS. 


IPO 


BY 

HORATIO  N.  ROBINSON,  LL.  D., 

LATH  PROFESSOR  Or  MATHEMATICS  IS  THE  UNITED  STATES  NAVT,   AXT>  A 


FULL  COCRSB   OF   MATIiEiLATlCS. 


NEW  YORK: 
IYISON,  PHINNEY  &  CO.,  48  AND  50  WALKER  ST. 

CHICAGO  :  S.  C.  GRIGGS  &  CO.,  39  AND  41  LAKE  ST. 

1804. 


ROBINSON'S  SERIES  OF  MATHEMATICS, 


The  most  COMPLETE,  most  PRACTICAL,  and  most  SCIENTIFIC  SE- 
RIES of  MATHEMATICAL  TEXT  BOOKS  ever  issued  in  this  country. 

(IN  TWENTY-TWO   VOLUMES.) 


1.  Robinson's  Progressive  Table-Book,  ....................... 

2.  Robinson's  Progressive  Primary  Arithmetic,  .............. 

3.  Robinson's  Progressive  Intellectual  Arithmetic,  .......... 

4.  Robinson's  Rudiments  ol  Written  Arithmetic,  ........... 

5.  Uobiiison's  Progressive  Practical  Arithmetic,  ............. 

6.  Robinson's  Key  to  Practical  Arithmetic,  .................... 

7.  Robinson'*  Progressive  Higher  Arithmetic,  ................ 

8.  Robinson's  Key  to  Higher  Arithmetic,  ...................... 

9.  Robinson's  New  Elementary  Algebra,  ........................ 

10.  Robinson's  Key  to  Elementary  Algebra,  .................... 

11.  Robinson'*  University  Algebra,  ........................... 

12.  Robiiisou'v  Key  to  University  Algebra,....  .................. 

13.  Robinson's  New  University  Algebra,  .......  .  ................. 

14.  Robinson's  Key  to  New  University  Algebra,  ...............  • 

15.  Robinaon's  New  Geometry  and  Trigonometry,  ............ 

16.  itobi'ison's  Surveying  and  Navigation,  ....................... 

17.  Robinson's  Analytical  Geometry  and  Conic  Sections,... 

18.  Robinson's  Din",  and  litteg.  Calculus,  (in  preparation)  ........ 

19.  u^biiison's  Elementary  Astronomy,  ....................... 

20.  'lobinson's  University  Astronomy,  .......................... 

21.  Wobinson's  Mathematical  Operations,    .......  .............. 

*•<>     Kobinson's  Key  to  Geometry  and  Trigonometry,  Conic 
Sections  and  Analytical  Geometry,  .......................... 


Entered,  according  to  Act  of  Congress,  in  the  year  1862,  by 
DANIEL  W.  FISH,  A.  M., 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Northern 
District  of  New  York. 

TRCAIR,  SMITH  A  MILES,  STEREOTYPERS,  STRACUSR.  N.  T. 


PREFACE. 


In  tlic  preparation  of  the  New  University  Algebra,  care  has  been  taken 
to  preserve  every  feature  of  the  original  work,  on  which  rested,  in  any 
degree,  its  claims  to  superiority.  The  aim  has  been  to  make  that  which 
was  good,  decidedly  better.  Hence  the  changes  that  have  been  made,  con- 
sist, for  the  most  part,  in  more  apt  arrangement,  in  large  additions  of  orig- 
inal matter,  and  in  presenting  the  whole  in  more  attractive  form. 

The  treatise,  as  now  submitted  to  the  public,  is,  indeed,  far  more  com- 
plete than  the  former,  not  only  in  the  range  of  topics,  but  also  in  gen- 
eral discussions  and  practical  applications.  In  many  parts  the  methods 
of  investigation  are  essentially  different, — the  object  being,  in  some  in- 
stances, to  secure  simplicity  in  logical  arrangement,  and  in  others,  to  es- 
tablish principles  and  rules  by  more  general  and  rigorous  demonstrations. 

The  articles  on  Inequalities,  Differential  Method  of  Series,  and  Interpo- 
Intior,  which,  in  the  old  treatise,  appear  as  an  appendix,  have  been  elabo- 
rated, and  made  to  take  their  appropriate  place  in  the  body  of  the  work. 

The  section  on  Radical  Quantities  is  quite  full,  embracing  the  more 
important  properties  of  Imaginary  Quantities  and  Quadratic  Surds,  be- 
sides a  complete  logical  development  of  the  Theory  of  Exponents. 

As,  in  the  author's  New  Elementary  Algebra,  the  Binomial  Theorem 

/ 

has  been  fully  investigated  with  reference  to  integral  exponents,  it  has 
been  deemed  unnecessary  to  repeat  here  the  particular  demonstration. 
Accordingly,  the  whole  subject  is  deferred  till  the  section  on  Series  is 
reached,  where  a  general  demonstration  of  this  theorem  is  given  in  a  con- 
cise way,  and  a  full  variety  of  applications  added.  The  whole  subject, 
as  presented  in  this  connection,  with  the  accompanying  illustrations,  can 
not  fail  to  interest  the  lovers  of  Algebra. 

The  General  Theory  of  Equations  is  treated  in  two  sections,  the  one 
embracing  the  general  properties  of  equations,  and  the  other  the  solution 


IV  PllEFACE. 

of  numerical  equations  of  all  degrees.  The  whole  subject  is  here  pre- 
sented, however,  in  a  condensed  form,  the  student  being  conducted,  in  a 
manner  direct  as  possible,  from  the  theoretical  to  the  practical.  The 
section  on  the  Properties  of  Equations,  it  is  proper  to  say,  owes  its  im- 
proved character  to  the  able  hand  of  Prof.  I.  F.  QUINIJY,  of  the  University 
of  Rochester,  whose  services  in  perfecting  other  books  of  this  Series  de- 
serves especial  mention. 

The  effort  which  has  been  made  in  this  treatise,  to  combine  the  lest 
practical  with  the  highest  theoretical  character,  is  specially  commended  to 
the  notice  of  the  true  educator.  Great  care  has  been  taken  everywhere 
to  set  forth  in  distinct  form  the  principles  of  the  science,  their  exact  log- 
ical relations  being  noted  by  proper  references ;  while  due  prominence  has 
been  given  to  those  numerous  precepts  and  expedients  which  are  so  nec- 
essary to  the  constitution  of  an  expert  Algebraist. 

The  design  throughout  has  been,  not  to  conceal,  but  fully  to  reveal  the 
difficulties  of  the  science,  and  to  encourage  the  learner,  not  to  avoid,  but 
to  grapple  with,  and  overcome  them ;  since,  to  the  student  of  Mathemat- 
ics, labor  rightly  directed,  is  discipline, — and  discipline,  after  all,  is  the  true 
end  of  education. 

It  is  but  just  to  state,  that  J.  C.  PORTER,  A.  M.,  has  had  the  constant 
care  and  supervision  of  the  present  work,  having  also  rendered  important 
assistance  in  the  preparation  of  some  other  works  of  the  Series, — a  fact 
which,  considering  his  long  and  distinguished  success  as  a  teacher  of 
Mathematics,  and  his  acknowledged  ability  as  a  mathematical  scholar, 
ought  to  afford  a  sufficient  guarantee*  for  the  utmost  accuracy  and  class- 
room fitness  on  every  page. 

Thus  distinguished  for  fullness  of  matter ;  for  scientific  arrangement; 
for  ample  discussion  ^nd  rigid  demonstration ;  for  clear  statement  and 
close  definition;  for  rules  brief  and  of  easy  application;  for  examples 
numerous,  apt  and  strictly  practical;  for  the  nicest  adaptation  to  the 
purposes  of  teaching ,  for  the  finest  mechanical  execution ;  for  whatever,  in 
short,  care,  skill,  science  and  taste  can  accomplish ; — the  New  University 
Algebra  is  submitted  to  the  public. 

July,  18G2. 


CONTENTS. 


SECTION  I. 

DEFINITIONS   AND    NOTATION. 

PAGE, 

General  Definitions 9 

Symbols  of  Quantity 10 

(symbols  of  Operation 10 

{symbols  of  Kelution 12 

Composition  of  Algebraic  Quantities T 14 

Axioms 15 

Exercises  in  Algebraic  Notation 16 

Computation  of  Numerical  Values - 17 

Signification  of  Plus  and  Minus  Signs 18 

ENTIRE    QUANTITIES. 

Addition 20 

Subtraction 25 

Multiplication 31 

Formulas  and  General  Principles 36 

Division 39 

Exact  Division 43 

General  Relations  in  Division 44 

Reciprocals,  Zero  Powers,  and  Negative  Exponents 45 

Divisibility  of  a™  ±  bm 46 

Greatest  Common  Divisor 52 

Least  Common  Multiple f 60 

FRACTIONS. 

Definitions  and  General  Principles 64 

Reduction 66 

Addition , 74 

Subtraction 76 

Multiplication v 77 

Division 79 

Reduction  of  Complex  Forms .-. : 81 

(v) 


VI  CONTENTS. 

SECTION  II. 

SIMPLE   EQUATIONS. 

Definitions 83 

Transformation  of  Equations. 85 

Reduction  of  Simple  Equations 89 

Problems 94 

Two  Unknown  Quantities 103 

Elimination 104 

Three  or  more  Unknowrn  Quantities 112 

Problems 118 

General  Solution  of  Problems 124 

Discussion  of  Problems 130 

Nothing  and  Infinity 135 

Interpretation  of  Anomalous  Forms 136 

Problem  of  the  Couriers 138 

Inequalities 145 

SECTION  III. 

POWERS   AND   ROOTS. 

Involution 151 

Powers  of  Monomials 151 

Powers  of  Fractions 154 

Discussion  of  Negative  Indices 155 

Powers  of  Polynomials 157 

Polynomial  Squares 158 

Evolution ^ ICO 

Roots  of  Monomials 161 

Square  Root  of  Polynomials 164 

Square  Root  of  Numbers 166 

Cube  Root  of  Polynomials 172 

Cube  Root  of  Numbers 176 

SECTION  IV. 

RADICAL   QUANTITIES. 

Reduction  of  Radicals 182 

Addition  of  Radicals 187 

Subtraction  of  Radicals 189 

Multiplication  of  Radicals IflO 

Division  of  Radicals 191 

Powers  and  Roots  of  Radicals 193 

General  Theory  of  Exponents 197 

Imaginary  Quantities : . .  .201 


CONTENTS.  Vll 

Properties  of  Quadratic  Surds 204 

Square  Root  of  Binomial  Surds 206 

Rationalization 208 

Radical  Equations 212 

SECTION  V.; 

QUADRATIC   EQUATIONS. 

Pure  Quadratics 216 

Affected  Quadratics 218 

Second  Method  of  completing  the  Square 221 

Treatment  of  Special  Cases * 224 

Equations  in  the  Quadratic  Form 228 

Examples  of  Equations  Solved  like  Quadratics 232 

Promiscuous  Examples  in  Quadratics 234 

Simultaneous  Equations  containing  Quadratics 236 

Examples  of  Simultaneous  Equations 243 

Theory  of  Quadratics 247 

Discussion  of  the  Four  Forms 250 

Discussion  of  Problems 252 

Interpretation  of  Imaginary  Results 253 

Problem  of  the  Lights 254 

Problems  producing  Quadratic  Equations 258 

SECTION   VI. 

\ 

PROPORTION,    PERMUTATIONS   AND    COMBINATIONS. 

Proportion 265 

Propositions  in  Proportion 267 

Problems  in  Proportion 274 

Permutations  and  Combinations 278 

Examples  of  Permutations  and  Combinations 283 

SECTION    VII. 

OF   SERIES. 

Arithmetical  Progression 285 

Application  of  the  Formulas 287 

The  Ten  Cases 290 

Problems 291 

^Geometrical  Progression 293 

Application  of  the  Formulas 294 

Problems 298 

Identical  Equations 301 

Decomposition  of  Rational  Fractions 306 


VI 11  CONTENTS. 

The  Residual  Formula 808 

Binomial  Theorem 310 

Application  of  the  Binomial  Formula 313 

Method  of  Substitution 317 

French's  Theorem 318 

Development  of  Surd  Roots  into  Scries 320 

Expansion  of  Fractions  into  Series 323 

Method  of  Indeterminate  Coefficients 325 

Reversion  of  Series 328 

Summation  of  Infinite  Series 331 

Recurring  Series % 332 

Differential  Method 336 

Interpolation 340 

Logarithms 343 

Properties  of  Logarithms 343 

Rules  for  Computation 345 

The  Common  System 34G 

Computation  of  Logarithms 347 

Use  of  Tables 353 

-Exponential  Equations 857 

SECTION  VIII, 

PROPERTIES   OP  EQUATIONS. 

General  Properties 359 

-  Commensurable  Roots 370 

__  Derived  Polynomials 374 

__  Composition  of  Derived  Polynomials 375 

-  Equal  Roots 370 

_  Transformations 380 

Detached  Coefficients 388 

^Synthetic  Division 392 

vSurd  and  Imaginary  Roots 398 

—  Rule  of  Des  Canes 400 

^.  Cardan's  Rule  for  Cubics .401 

SECTION  IX. 

SOLUTION    OF   NUMERICAL   EQUATIONS   OP   HIGHER   DEGREES. 
_-  Limits  of  Roots :405 

—  Limiting  Equation 408 

—  Sturm's  Theorem 410 

.    Homer's  Method  of  Approximation 416 


A  TREATISE  ON  ALGEBRA. 


SECTION  I. 
DEFINITIONS  AND  NOTATION. 

1.  Quantity  is  anything  that  can  be  increased,  diminished,  or 
measured  ;  as  distance,  space,  weight,  motion,  time. 

A  quantity  is  measured  by  finding  how  many  times  it  contains  a 
certain  other  quantity  of  the  same  kind,  regarded  as  a  standard. 
The  conventional  standard  thus  used  is  called  the  wilt  of  measure. 

•£.  Mathematics  is  the  scieuce  which  treats  of  the  properties 
and  relations  of  quantities.  It  employs  a  peculiar  language,  con- 
sisting of  symbols,  to  express  the  values  of  quantities,  and  the 
operations  to  which  these  values  are  subjected.  The  symbols  are 
of  three  kinds,  as  follows  : — 

1st.  Symbols  of  Quantity,  consisting  of  figures  or  numerals  used 
in  arithmetical  computations,  letters  and  other  characters  used  in 
general  analysis,  and  graphic  representations  or  drawings  used  in 
geometrical  investigations. 

lid.  Symbols  of  Operation,  consisting  of  the  signs  or  characters 
employed  to  indicate  those  mathematical  processes  by  which  quanti- 
ties are  made  to  undergo  changes  of  value,  such  as  addition,  sub- 
'traction,  multiplication  and  division. 

3d.  Symbols  of  Relation,  consisting  ot  the  signs  used  in  com- 
paring quantities  with  respect  to  their  relative  magnitudes,  and 
certain  abbreviations  employed  in  the  process  of  reasoning. 
•—  3.  Algebra  is  that  branch  of  mathematics  iu  which  quantities 
are  represented  by  letters,  and  the  operations  arid  relations  are 
indicated  by  signs.  The  object  of  algebraic  notation  is  to  abridge 
and  generalize  the  analysis  of  mathematical  problems.  Algebra  is 
therefore  a  species  of  universal  arithmetic. 

(9) 


10  ALGEBRAIC    .QUANTITIES 

SYMBOLS    OF    QUANTITY. 

4.  An  Algebraic  Quantity  is  a  quantity  expressed  in  algebraic 
language.  There  are  two  kinds  of  algebraic  quantities — known  and 
unknown. 

o.  Known  Quantities  are  those  whose  values  are  given ;  when 
these  are  not  expressed  l>y  figures  they  are  represented  by  the 
leading  letters  of  the  alphabet,  as  a,  b,  c,  d. 

©.  Unknown  Quantities  are  those  whose  values  are  to  be  deter- 
mined ;  they  are  represented  by  the  final  letters  of  the  alphabet,  as 
u,  x,  y,  z. 

f .  The  small  italic  letters  just  given  are  the  more  common  sym- 
bols of  quantity.  In  addition  to  these,  capital  letters  are  sometimes 
employed,  as  A,  B,  6',  D,  X,  Y,  Z,  etc. 

Quantities  which  have  like  relations  to  a  series  of  quantities  in 
any  investigation,  are  sometimes  represented  by  a  single  letter 
repeated  with  different  accents,  as  a,  a',  a",  a'"  a"" ,  read,  a,  a 
prime,  a  second,  a  third,  etc.;  or  by  a  letter  repeated  with  different 
subscript  figures,  as  a,  a,,  «3,  a3,  a4,  etc.,  read,  a,  a  sub  one, a  sub 
two,  a  sub  three,  etc. 

In  certain  investigations  it  is  convenient  to  represent  quantities 
by  the  initial  letters  of  their  names.  Thus,  S  or  s  may  represent 
sum  ;  D  or  <f,  difference  or  diameter;  R  or  r,  ratio,  remainder  or  ra- 
dim.  In  some  cases  the  capital  and  the  small  letter  may  be  used  to- 
gether to  distinguish  between  two  quantities  of  the  same  kind.  Thus, 
in  a  problem  relating  to  two  circles,  r  may  represent  the  radius  of 
the  smaller,  and  R  the  radius  of  the  larger  circle. 

SYMBOLS     OF    OPERATION. 

8.  The  Sign  of  Addition  is  the  perpendicular  cross,  -f->  called 
plus.     It  indicates  that  the  quantity  written  after  it  is  to  be  added 
to  the  other  quantity  or  quantities   in  the  expression.     Thus,  in 
a-{-b,  the  sign  indicates  that  the  quantity  b  is  to  be  added  to  the 
quantity  a;  and  the  expression  is  read,  a  plus  b. 

9.  The    Sign  of  Subtraction  is    a    short    horizontal    line,  — , 
called  minus.     It  indicates  that  the  quantity  written  after  it  is  to  bo 
subtracted  irom  the  other  quantity  or  quantities  in  the  expression. 


DEFINITIONS   AND    NOTATION.  11 

Thus,  in  a — b,  or  — 6-j-a,the  minus  sign  indicates  that  the  quantity 
I  is  to  be  subtracted  from  the  quantity  a ;  and  the  expression  is 
read,  a  minus  b,  or  minus  b  plus  <(. 

The  sign  —  may  be  written  between  two  quantities  to  indicate 
that  their  arithmetical  difference  is  to  be  takejp,  when  it  is  not  known 
which  is  the  greater. 

10.  The  Double  Sign,    ±,  is  written  before  a  quantity  to  indi- 
cate that  it  is  to  be  both  added  and  subtracted  ;  it  serves  to  unite 
in  a  single  expression   two  combinations   of  the  same   quantities. 
Thus,  a±b  is  equivalent  to  a-^-b  and  a — b,  and  is  read  a  plus  01 
minus  b. 

11.  The  Sign  of  Multiplication  is  the  oblique  cross,  x.     It  in- 
dicates that  the  quantity  before  it  is  to  be  multiplied  by  the  quantity 
after  it.     Thus,  in  ax£>,  the  sign  indicates  that  a  is  to  be  multiplied 
by  b.     Instead  of  the  sign,  x,  a  point  is  sometimes  used  to  denote 
multiplication  ;  as  3 >'xmy,  which  signifies  the  same  as  o  x  x  x,y. 

The  multiplication  of  quantities  which  are  represented  by  letters, 
is  generally  indicated  by  writing  the  factors  one  after  another  with- 
out any  intervening  sign.  Thus,  3abc  signifies  the  same  as 
3  xa  x  b  x  c,  or  3-a-b-c.  It  is  evident  that  this  notation  cannot  be 
employed  when  the  several  factors  are  represented  by  figures.  We 
cannot  represent  3  times  4  by  simply  writing  the  factors  together, 
thus,  3  4  ',  for  the  product  thus  indicated  could  not  be  distinguished 
from  the  number  34. 

NOTES.  1.  The  result  of  any  multiplication  is  called  a  product,  and  the 
quantities  multiplied  are  called  factors. 

2.  When  the  quantities  to  be  multiplied  are  represented  by  letters, 
they  are  called  literal  factors  ;  when  they  are  represented  by  figures, 
they  are  called  numerical  factors. 

12.  The  Sign  of  Division  is  a  short   horizontal  line  with  a 
point  above  and  one  below,  -^.     It  indicates   that  the   quantity 
before  it  is  to  be  divided  by  the  quantity  after  it.     Thus,  in  a-±-b, 
the  sign  indicates  that  a  is  to  be  divided  by  b. 

Division  is  also  expressed  by  writing  the  dividend  above,  and  the 

divisor  below,  a  short  horizontal   line  ;  as  1. 

b 

13.  The  Sign  of  Involution  is  a  number  written  above  and  to 
the  right  of  a  quantity,  to  indicate  how  many  times  the  quantity  is 


12.  ALGEBRAIC    QUANTITIES. 

to  be  taken  as  a  factor.  Thus,  in  a6,  the  number  5  indicates  that 
a  is  to  be  taken  5  times  as  a  factor;  and  the  expression  is  equivalent 
to  aaaaa. 

A  factor  repeated  to  form  a  product  is  called  a  root }  the  product 
itself  is  called  a  power  ;  and  the  figure  which  indicates  how  many 
times  the  root  or  factor  is  taken,  is  called  the  exponent  of  the  pow- 
er. Thus,  in  the  indicated  product  a6,  a  is  the  root,  a5  is  the  power, 
called  the  5th  power  of  a,  and  5  is  the  exponent  of  this  power. 
When  no  exponent  is  written  over  a  quantity,  the  exponent  1 
may  always  be  understood. 

NOTE. — For  the  sake  of  brevity,  the  exponent  of  the  power  may  be 
called  the  exponent  of  the  letter  or  quantity  over  which  it  is  placed. 
Thus  in  «5,  5  may  be  called  the  exponent  of  a. 

14.  The  Sign  of  Evolution,  or  Radical  Sign,  is  the  character 
•\/ .  It  indicates  that  some  root  of  the  quantity  after  it  is  to  be  ex- 
tracted. The  name  or  index  of  the  required  root  is  the  number 
written  above  the  radical  sign.  Thus,  \/a  denotes  the  cube  root  of 
a]  Va  denotes  the  4th  root  of  a;  and  so  on.  When  no  index  is 
written  over  the  sign,  the  index  2  is  understood ;  thus,  v/«  denotes 
the  square  root  of  a. 

Fractional  Exponents  are  also  used  as  the  sign  of  evolution,  the 
denominator  being  the  index  of  the  required  root.  Thus,  in  aa, 
the  denominator,  3,  indicates  that  the  cube  root  of  a.  is  required, 
and  the  expression  is  equivalent  to  V«- 

Fractional  exponents  are  used  to  denote  both  involution  and 
evolution  in  the  same  expression,  the  numerator  indicating  the 
power  to  which  the  quantity  is  to  be  raised,  and  the  denominator 
the  required  root  of  this  power.  Thus,  the  expression  a*  signifies 
the  4th  root  of  the  3d  power  of  a,  and  is  equivalent  to  V«s. 


SYMBOLS   OF   RELATION. 

15.  The  Sign  of  Equality  is  two  short  horizontal  lines,  —. 
It  indicates  that  the  two  quantities  between  which  it  is  placed  are 
equal.  Thus,  in  a=u-±-c,  the  sign,  ==,  indicates  that  a  is  equal  to 
b  plus  <;.  An  expression  of  equality  between  two  quantities  is 
called  an  equal '<JH. 


DEFINITIONS    AND    NOTATION. 

16.  The  Sign  of  Inequality  is  the  angle,  > .     It  indicates  that 
the  quantities  between  which  it  is  written  are  unequal,  the  opening 
being  always   turned   toward  the  greater.     When   the  opening  is 
toward  the  left,  it  is  read  greater  than  ;  wher  the  point  or  vertex 
is  toward  the  left,  it  is  read  less  than.     T^us,  a>b  signifies  that  a 
is  greater  than  b  ;  x+y<z  signifies  that  x  plus  y  is  less  than  z. 

17.  The  Signs  of  Aggregation  are  the  parenthesis,  (),  brackets, 

[  ],  brace,  j    I,  vinculum, ,  and  bar,  |  .     They  indicate  that 

the  quantities  included  within,  or  connected  by  them,  are  to  be  ta- 
ken collectively  and  subjected  to  the  same  operation.     Thus, 

(a+b — c)  x,     [a-\-b — c]  x,     {a+b — c}ar,  a-f  b— c  x  x, 


+a 
and    -\-b 


x 


are   expressions  signifying    that  the  whole  quantity, 


a-\-b — c,  is  to  be  multiplied  by  x.  Two  or  more  of  these  signs 
may  be  used  correlativcly  in  the  same  -expression,  in  which  case1  the 
brackets  should  include  the  parenthesis  or  vinculum,  and  the  brace 
should  include  the  brackets ;  thus, 

jm— «[c— &(m+rf)]+3}. 

18.  The  Sign  of  Continuation  is  a  succession  of  points,  indicating 
that  a  series  of  quantities  may  be  continued  indefinitely  according  to 

the  same  law.     Thus,  in  the  expression,  a+ a2 -\-a3+a4-{- , 

the  points  indicate  that  the  series  has  an  infinite  number  of  terms, 
all  formed  according  to  the  same  law. 

19.  The  Sign  of  Katio  is  two  points  like  the  colon,  :  ,  placed 
between  the  quantities  compared.     Thus,  the  expression,  a  :  b,  sig- 
nifies the  ratio  of  a  to  b. 

SO.  The  Sign  of  Proportion  is  a  combination  of  the  sign  of 
ratio  and  the  sign  of  equality,  :  =  :  ;  or  a  combination  of  points 
only,  :  ::  :  .  Thus,  a  :  b=c  :  d,  signifies  that  the  ratio  of  a  to  b 
is  equal  to  the  ratio  of  c  to  d ;  and  the  expression  a  :  b::  c  :  d 
signifies  the  same,  and  may  be  read,  a  is  to  b  as  c  is  to  d. 

21.  The  Sign  of  Variation  is  the  character  oc  .  It  signifies 
that  the  two  quantities  between  -which  it  is  placed,  whether  equal  or 
unequal,  increase  or  diminish  together,  so  as  to  preserve  constantly 
the  same  ratio. 

NOTE. — The  signs  of  ratio,  proportion,  and  variation,  will  be  more  ful- 
ly explained  hereafter. 
2 


14  ALGEBRAIC    QUANTITIES. 

COMPOSITION    OP   ALGEBRAIC    QUANTITIES. 


An  algebraic  quantity  may  consist  of  a  single  letter  or 
element,  or  a  combination  of  symbols  as  factors,  or  several  combina- 
tions or  parts.  The  parts  are  called  terms  ;  hence, 

22.  The  Terms  of  an  algebraic  quantity  are  the  parts  or  divis- 
ions made  by  the  signs  -f-  a]Qd  —  .  Thus,  in  the  quantity 
5a-}--&2  —  ex,  there  are  three  terms,  of  which  5a  is  the  first,  +'lb'2 
the  second,  and  —  ex  the  third. 

2&.  When  a  quantity  consists  of  a  single  term,  it  is  said  to  be 
simple  ;  when  it  is  composed  of  two  or  more  terms,  it  is  said  to  be 
compound. 

25.  Positive  Terms  are  those  which  have   the  plus  sign  •  as 
-\-x,  or  -j-2eV.     The  first  term  of  an  algebraic  quantity,  if  written 
without  any  sign,  is  positive,  the  plus  sign  being  understood. 

26.  Negative  Terms  are  those  which  have  the  minus  sign  ;  as 
—  3a,  or  —  2m.x2.     The  sign  of  a  negative  quantity  is  never  omitted. 

27.  A  Coefficient  is  a  number  or  quantity  prefixed  to  another 
quantity,  to  denote  how  many  times  the  latter  is  taken.     Thus,  in 
3.r,   the  number  8   is   the  coefficient  of  x,  and  indicates  that  x  is 
taken  8  times;  hence,  the  expression  8,r  is  equivalent  to  x  -\-x-\-x. 
In  4ax,  4  may  be  regarded  as  the  coefficient  of  ax,  or  4  a  as  the 
coefficient  of  x.     In  5(a+:r),  5  is  the  coefficient  of  a+x.     When 
no  coefficient  is  written,  the  unit  1  is  understood. 

2&.  It  should  be  observed  that  in  a  term  having  the  plus  sign, 
the  coefficient  shows  how  many  times  the  quantity  is  taken  additivc- 
ly  ;  and  in  a  term  having  the  minus  sign,  the  coefficient  shows  how 
many  times  the  quantity  is  taken  mltr  actively.     Thus, 
-|_3a=+a-|-a+a 
—  3a  =  —  a  —  a  —  a 

29,  Similar  Terms  are  terms  containing  the  same  letters, 
affected  with  the  same  exponents  ;  the  signs  and  coefficients  may 
differ,  and  the  terms  still  be  similar.  Thus,  3x2  and  7.ra  are  similar* 
terms  ;  also,  2m^/2  and  —  bmd?  are  similar  terms. 

«£O.  Dissimilar  Terms  are  those  which  have  different  letters  or 
exponents.  Thus,  axy  and  ayz  are  dissimilar;  also  3ary  and  -'>x*i/*. 

31.  A  Monomial  is  an  algebraic  quantity  consisting  of  only  ono 
term  ;  as  3x,  or  —  7xy. 


DEFINITIONS   AND   NOTATION.  15 

.  A  Polynomial  is  an  algebraic  quantity  consisting  of  more 
than  one  term;  as  x-\-y,  or  4a2— 3x-\-m. 

33.  A  Binomial   is  a  polynomial  of  two  terms;  as  a-{-b,   or 
3*— z. 

34.  A  Residual  is  a  binomial,  the  two  terms  of  which  are  con- 
nected by  the  minus  sign  ;  as  a— b,  or  4.r — 3y. 

3«>.  A  Trinomial  is  a  polynomial  of  three  terms;  as  x+y+z, 
or  la— W+d. 

£»G.  The  Degree  of  a  term  is  the  number  of  its  literal  factors. 
Since  the  exponents  show  how  many  times  the  different  letters  are 
taken  as  factors,  the  degree  of  a  term  is  always  found  by  adding 
the  exponents  of  all  the  letters.  Thus,  x  and  5y  are  terms  of  the 
first  degree  ;  a2  and  4a&  are  terms  of  the  second  degree  ;  x3,  3j2iy, 
o.ry2,  and  ±xyz  are  terms  of  the  third  degree. 

37.  A  Homogeneous  Quantity  is  one  whose  terms  arc  all  of 
the  same  degree  ;  as  x3 — bx*y-\-%xyz. 

38.  A  Function  of  a   quantity  is  any  expression   containing 
that  quantity.     Thus  ax*  is  a  function  of  x'f  3y-|-%— 4  is  a  func- 
tion of  y. 

AXIOMS. 

3O.  An  Axiom  is  a  self-evident  truth.  The  following  axioms 
underlie  the  principles  of  all  algebraic  operations  : 

1.  If  the  same    quantity  or  equal  quantities  be  added  to  equal 
quantities,  the  sums  will  be  equal. 

2.  If  the  same  quantity  or  equal  quantities  be  subtracted  from 
equal  quantities,  the  remainders  will  be  equal. 

;•>.  If  equal  quantities  be  multiplied  by  the  same,  or  equal  quanti- 
ties, the  products  will  be  equal. 

'4.  If  equal  quantities  be  divided  by  the  same,  or  equal  quantities, 
the  quotients  will  be  equal. 

5.  If  a  quantity  be  both  increased  and  diminished  by  another,  its 
value  will  not  be  changed. 

B.  If  a  quantity  be  both  multiplied  and  divided  by  another,  its 
value  will  not  be  changed. 

7.  Quantities  which  are  respectively  equal  to  the  same  quantity, 
are  equal  to  each  other. 


16  ALGEBRAIC    QUANTITIES. 

8.  Like  powers  of  equal  quantities  are  equal. 

9.  Like  roots  of  equal  quantities  are  equal. 

10.  The  whole  of  a  quantity  is  greater  than  any  of  its  parts. 

11.  The  whole  of  a  quantity  is  equal  to  the  sum  of  all  its  parts. 


EXERCISES  IN    ALGEBRAIC  NOTATION. 

4O.  In  the  examples  which  follow,  it  is  required  of  the  pupil 
simply  to  express  given  relations  in  algebraic  language. 

1 .  Give  the  algebraic  expression  for  the  square  of  a  increased  by 
by  4  times  b.  Ans.  aa-j-4&. 

2.  Give  the  algebraic  expression  for  7  times  the  product  of  x  and 
y,  diminished  by  5  times  the  cube  of  z. 

8.  Indicate  the  quotient  of  12  times  the  square  of  a  minus  5  times 
the  cube  of  b,  divided  by  the  sum  of  a  and  c. 

4.  If  d  represent  a  person's  daily  wages,  what  will  represent  his 
wages  for  6  days  ?  Am.  Qd. 

5.  An  army  drawn  up  in  rectangular  form,  has  b  men  in  rank, 
and  a  men  in  file ;  of  how  many  men  is  the  army  composed  ? 

6.  If  a  man  labor  m  days  in  a  week  at  c  dollars  per  day,  what 
will  his  earnings  amount  to  in  7  weeks  ? 

7.  The  length  of  a  prism  is  a,  the  breadth  c,  and  the  altitude 
a— c;  required  the  solid  contents.  Ans.  ac(a — c). 

8.  A  has  4m  dollars,  B  has  m  times  as  many  dollars  as  A,  and  C 
has  3  times  as  many  dollars  as  B  wanting  d  dollars;    how  many 
dollars  has  C  ? 

9.  A  dealer  sells  b  sheep  and  c  calves,  at  an  average  price  of  m 
dollars  per  head ;  how  much  does  he  receive  for  all  ? 

10.  A  man  has  3  square  lots  measuring  m  rods  on  a  side ;  how 
many  acres  in  the  3  lots  ?  .        3ma 

'  160* 

1 1 .  From  a  rectangular  piece  of  land  whose  length  was  a  rods 
and  whose  width  was  b  rods,  there  were  sold  c  acres ;  how  many 
acres  remained  unsold  ? 

12.  A  ship  laden  with  a  barrels  of  flour,  valued  at  m  dollars  per 


DEFINITIONS    AND    .NOTATION.  17 

barrel,  met  with  a  disaster  by  which  I  barrels  were  lost,  and  the 
remainder  damaged  to  the  amount  of  d  dollars  per  barrel  \  what  was 
the  worth  of  the  remainder  ?  Ans.  (a — U)(in — d). 

13.  A  man  having  c  acres  of  land  worth  £  dollars  per  acre,  divi- 
ded its  value  equally  between  m  sons  and  one  daughter)  how  many 
dollars  did  each  receive  1 

14.  A  company  of  n  persons  began  business  with  a  joint  capital 
of  c  dollar*.     The  first  year  they  gained  b  dollars,  the  second  year 
they  lost  d  dollars,  the  third  year  they  doubled   the   capital  with 
which  they  began  that  year,  and  then  dissolved  partnership,  sharing 
equally  their  accumulated  capital ;  what  was  each  man's  share  ? 


COMPUTATION    OF    NUMERICAL    VALUES. 

41.  The  Numerical  Value  of  an  algebraic  quantity  is  the  num- 
ber obtained  by  assigning  numerical  values  to  all  the  letters,  and 
performing  the  operations  indicated. 

1.  Whit  is  the  numerical  value  of  (a2—  bc)a,  when  a=30, 
6=25,  and  c=2S  ? 

OPERATION. 

(a3—  bc)a=  (30x30—  25x28)  x  30  =200x30  =  6000,  Ans. 
Find  the  numerical  values  of  the  following  expressions,  in  which 
0=12;  £=10;  c=8;  m=G'}  w=5;  d=2. 

2.  a'—  be.  Ans.  64. 

3.  (a+ld)m.  Ans.  192. 

4.  ani-f-c2  —  md*.  Ans.  40. 

5.  (a-f  6)m—  (c+</>.  Ans.  82. 
G.  [4a2-(36a—  2c)]J.                                               Ans.  584. 

7.  (a2—  i)  (&'—  a).  Ans.  11792. 

4  a  —  c 

Ans.  8. 


3flm_-(6»+2c)     a1—  d* 

_  -^  —  '  _  -±.  __  Ans.  24. 

26+n 

2*     , 


18  ALGEBRAIC    QUANTITIES. 

Find  the  numerical  values  of  the  following  expressions,  in  which 
a  =  S  ;  b=G  ;  c=4  ;  d=2  ;  m=3  ;  n=l. 


______ 

11          '       '    _Z  __  .  Ans.  6. 

'    a-j-  b  -fc 


12.    (6asvi—  4mV)  (m'-7w).  ^Lns.  336. 

13  (5a»+26X  ^  11. 


15.    |2c(aV—  m8)  —  2(56'+4m)  -j-cjd1.  -4ns.  3424. 


1  /am9      t?4—  1 

- 
m 


17.    -    -  --h-i-r 

\   c     '   wa  — 


18.    -     [a-|-2cxw^—  c?]»i—  2  (ai-f-m')     * 


19. 
20 


fa  b    \       /     a  5\  >!! 

21.      -TT-f  —  itl-Hhl  —  7  --  TT)-  -4m.  1. 

\a-f-6  '  a  —  &/       \a  —  b     a--lJ 


-\-l 

6a3-22a-f-18 

__  '  -4ns. 

a»_6a2-fll«—  6 


.  1T!U. 


SIGNIFICATION    OF   THE   PLUS   AND    MINUS    SIGNS. 

.  The  signs,  -\-  and  — ,  have  heen  denned  as  symbols  of 
operation,  the  former  indicating  addition,  and  the  latter  subtraction. 
Now  when  we  meet  with  detached  or  single  terms  affected  with  the 
plus  or  minus  signs,  as  for  instance  in  the  examples  of  addition, 
subtraction,  multiplication  or  division,  we  are  to  consider  the  positive 


DEFINITIONS    AND    NOTATION.  19 

terms  as  quantities  taken  adtlitwely,  and  the  negative  terms  as 
quantities  taken  suhtr actively  ;  and  this  is  the  only  signification  that 
need  be  attached  to  these  signs,  at  present. 

It  is  obvious  that  positive  and  negative  terms,  as  just  explained, 
are  in  an  important  sense  opposites.  And  *it  will  be  shown  in  a 
future  section  (182  ^  that  quantities  sustaining  to  each  other  various 
other  relations  of  opposition  or  contrariety,  are  distinguished  by  the 
plus  and  minus  signs. 

4:3.  In  order  to  establish  general  rules  for  algebraic  operations, 
it  will  be  necessary  in  this  place  to  recognize  the  following  principles, 
consequent  upon  the  peculiar  manner  of  considering  quantities  in 
Algebra : 

1. — A  quantity  may  be  considered  as  having  two  kinds  of 
value, — an  absolute  or  numerical  value  determined  simply  by  the 
number  of  units  it  contains,  and  an  algebraic  value  depending  on 
the  sign. 

2.— Two  quantities  having  the  same  absolute  value,  but  affected 
with  unlike  signs,  are  not  algebraically  equal.  Thus,  5a  is  not 
equal  to  — 5«,  for  the  former  expression  signifies  that  a  is  taken 
additively  five  times,  and  the  latter  signifies  that  a  is  taken  subtract- 
ively  five  times. 

3. — Two  quantities  having  the  same  absolute  value,  but  affected 
with  unlike  signs,  are  together  equal  to  zero.  Thus  if  a  denote  tho 
absolute  value  of  any  quantity,  then 

-fa— a=0 
-f2a— 2a=0 

=0,etc. 


20  ENTIRE   QUANTITIES. 


ADDITION. 

44.  Addition,  in  Algebra,  is  the  process  of  uniting  two  or  more 
quantities  into  one  equivalent  expression  called  their  sum. 

4«>.  The  term,  addition,  has  a  more  general  meaning  in  Algebra 
than  in  Arithmetic,  because  the  quantities  to  be  added  may  be  either 
positive  or  negative. 

4G.  The  Arithmetical  Sum  of  two  or  more  quantities  is  the 
sum  of  their  absolute  values,  and  has  reference  simply  to  the  num- 
ber of  units  in  the  quantities  added. 

47.  The  Algebraic  Sum  of  two  or  more  quantities  is  a  quantity, 
which,  taken  with  reference   to  its  sign,  is  equivalent  to  the  given 
quantities,  each  taken  with  reference  to  its  particular  sign. 

48.  To  deduce  a  rule  for  addition,  which  will  conform  to  the 
nature  of  positive  and  negative  quantities,  let  us  consider  the  fol- 
lowing examples : 

1.  Add  4«,  3«,  and  5«. 

Since  in  these  quantities  a  is  taken  additwely,  4,  3,  and  5  times, 
or  12  times,  the  algebraic  sum  required  must  be  -|-12a;  or  simply, 
12a.  That  is, 

4a-f3a-f-5a=12a 

2.  Add  — 4a,  — 3a,  and  — 5«. 

Since  in  these  qantities  a  is  taken  subtr  actively ,  4,  3,  and  5  times, 
or  12  times,  the  algebraic  sum  required  must  be  — 12a.     That  is, 
— 4tt — 3a_  5a— — 12a 

Hence, 

The  algebraic  sum  of  two  or  more  similar  terms  having  like  signs, 
is  the  sum  of  their  absolute  values  taken  with  their  common  sign. 

3.  Add  la  and  —3 a. 
From  Ax.  11  we  have 

7o=4<r-f3a 

The  sum  of  la  and  — 3«  is  therefore  the  same  as  the  sum  of 
4a,  3« ,  and  — 3a.  But  since  3a  and  — 3a  taken  together  are  equal 
to  nothing,  (4:3,  3),  the  required  sum  must  be  the  remaining  term, 
4a.  That  is 

a)=4a-f  3a — 3a=4a,  Ans. 


ADDITION. 

4.  Add  — la  and  3«. 

From  Ax.  11, 

— la= — 4« — 3a 

The  sum  of  — la  and  3a  is  therefore  the  samc^alTtEio  sum  oi 
— 4« ,  — 3r?,  and  3«.  But,  since  — 3a  and  -J-oa  taken  together  arc 
equal  to  nothing,  (43?  3),  the  required  sum  must  be  the  remaining 
term,  — 4a.  That  is, 

— 7«-|-3«=— 4a — 3a-|-3a= — 4«,  Ans. 

Hence, 

The  algebraic  sum  of  two  similar  terms  having  unlike  signs,  is  the 
difference  of  their  absolute,  whit's  tfken  with  the  sign  of  the  greater 
term. 

It  may  further  be  observed, 

1st.  That  three  or  more  similar  terms,  having  different  signs,  may 
be  added,  by  first  finding  the  sum  of  the  positive  terms,  and  the  sum 
of  the  negative  terms,  separately,  and  then  adding  these  results. 
Thus, 

3a— 5<7— 4a+2a-|-8a=13a — 9a=4a 

2d.  Dissimilar  terms  cannot  be  united  into  one  term  by  addition, 
because  the  quantities  have  not  a  common  unit.  We  can  there- 
fore only  indicate  the  addition  of  dissimilar  terms,  by  connecting 
them  by  their  respective  signs.  Thus,  the  sum  of  a,  b,  and  — c  is 

a-\-b — c 

3d.  It  is  indifferent  in  what  order  the  terms  of  an  algebraic 
quantity  arc  written,  the  value  being  the  same  so  long  as  the  signs 
of  the  terms  remain  unchanged.  Thus, 

a-\-b — c=b-\-a — c=a — c-\-b=  — c-\-a-\-b 

For,  each  of  these  expressions  denotes  the  sum  of  the  three  terms, 
a,  h,  and  — c. 

49.  From  these  principles  and  illustrations  we  deduce  the 
following 

ROLE.     To  add  similar  terms  : 

I.  When  the  signs  are  alike,  add  the  coefficients,  and  prefix  the 
sum,  with  the  given  sign,  to  the  common  literal  part. 

II.  When  the  signs  are  unlike,  find  the  sum  of  the  positive  and 
of  the  negative  coefficients  separately,  and  prefix  the  difference  of 
the  two  sums,  with  the  sign  of  the  greater,  to  the  common  literal 
part. 


22  ENTIRE    QUANTITIES. 

To  add  polynomials  : 

I.  Write  the  quantities  to   be  added,  placing  the  similar  terms 
together  in  separate  columns. 

II.  Add  each  column,  and  connect  the  several  results  by  their 
respective  signs. 


EXAMPLES    FOR    PRACTICE. 

(1.)  (2.)  (3.) 

-{-  a*cx 


4xy  —  a'b  -}-Qa*cx 

l±xy 


(5.)  (6.) 

4z'— -3xy  — -Ta'c-f  m 

x*+2xy  -f4a'c— 3m 

4.  a*c — 2m 


I  4a8c-f5»i 

(8.)  (9.) 


4'c—  2a)—  m-f  4  5(a—  x9)-f3l/a—  x-f  5 

3(c_2a)-f  4?n—  8  4(a—  xa)—  2l/a^x+  8 

_  8(r—  2a)—  3m+12  2(o—  x2)—  81/JJHg—  12 

12(c—  2a-     m—  16  _ 


ll(c—  2a)-f-  ?7i—  8  10(a—  xa)—  5l/ 

10.  Add  12a2x,  5aa*,  —  4aarc,  6a2u,  and  —  10aax. 

Ans.  9aV. 

11.  Add  4ai^,  —2abd>,  labd*,  abd*,  —  5a£da,  —  ISaid*,  and 
7  aid*.  Ans.  —  oW. 

12.  Add  2x^—  2a2,  3a*+2xy,  a'-f  x,y,  4a2—  3xy,  and  2xy—2a*. 

Ans.  4rt*-|~4ay. 

13.  Add    8aV  —  3./y,    5ax  —  5^y,    9.ry—  5rjKc,    2aaxa-{-xy,    and 

Ans.   lOaV  —  a 


ADDITION. 


14.  Add  a'—2ac+cd+b,  ?>«*—Sac—3cd—2b,  2af+  ac 

6b,  and  a2  —  4ac-j-2cd—  36.  -4ns.  7  a2—  Sac—  5cd+2Z>. 

15.  Add   2aV—  3mx-f-4ma<:?,  3mad-f5aV  —  5mx,  6mx  —  4w3d 
_  3aV,  and  2mx  —  3«V—  -Sm'e?.  Ans.  aV. 

16.  Add  2bx—  12,  3z»—  26*,  5*1—  31/ay31/x+12,  and  xa+3. 

4ns.  9za-f  3. 

17.  Add  106'—  36xa,  2ft  V—61,  10—  26xa,  />V—  20,  and  36xa+ 

—  10. 


18.  Add  96c'—  18aca,  155c3H-ac,  9ac9—  246cs,  and  9aca—  2. 

Ans.  ac  —  2. 

19.  Add  Gm'+Som+l,  6aw^—  2ma-f-4,  2m1—  8aw-f-7,  and  3ma 

.  9m2 


20.  Add  5x4—  3x»-f4x»—  2.r-f  10,  7x4+2xf+2xf4-  5x+2,  and 

Ans.  12r4-f6xa-f  12. 

21.  Add   3*y—  5xV—  xV—  ^- 

,  and  aty1—  ay>-2x 


22.  Add  5a-j-3l/^a^i4-4,  7a—  T/V—  1—  5,  3a—  5l/ma—  1 
—8,  and  2a-f  2l/m2^14-2.  4ns.  17a—  T/m2—  1—  7. 

23.  What  is  the  sum   of  3aV—  2caa^+  a^c^,  2aac2-|-3caa^— 


24.     What  is  the  sum  of  9a(o  —  6)  —  4?7iV  m  —  c,  7ml/ 
6a(a—  6),  and  12  mVm^c—  8a(a—  &)  ? 


m  —  c  — 


25.  What  is  the  sum  of  a-\-b  -\-c-\-d-\-m,  a-\-b-{-c-\-d  —  m,  CT-|- 
6-|-c  —  d  —  m,  a-|~^  —  c  —  ^  —  m>  an(^  a  —  ^  —  c  —  ^  —  Wl  ? 

4ns.  fta-|-36-|-f—  e?—  3m. 

5O.  The  Unit  of  Addition  is  the  letter  or  quantity  whose  co- 
efficients are  added,  in  the  operation  of  finding  the  sum  of  two  or 
more  quantities.  Thus,  in  the  example, 


the  letter  x  is  the  unit  of  addition.     Also,  in  the  example, 


the  quantity,  V^cr+c,  is  the  unit  of  addition 


24  ENTIEE    QUANTITIES. 

•51.  When  dissimilar  terms  have  a  common  literal  part,  this  may 
be  taken  as  the  unit  of  addition.  The  sum  of  the  terms  will  then  be 
expressed  by  inclosing  the  sum  of  the  coefficients  in  a  parenthesis, 
and  prefixing  it  to  the  common  unit. 

EXAMPLES    FOR    PRACTICE. 

(2.)  (3.) 

\laxy*  (5a — 

(2c— 


(12a+2m).ry1  (la+c 

(5.) 
(  «'—  Si)  O9—  1) 


(3a  +3/Q  (ma- 
(a  ^/,-c)  (x«— 

6.  Add  ax,  2cx,  and  4c?x.  Ans.  (a+  2 

7.  Add  '  y-j-^?  3flry-j-2ca;,  and 


,  and  (a+fyx+Zcdxy. 
Ans. 
9.  Add  rar-j-Ty,  Tax  —  oy,  and  — 

-4ns.  (8a  —  Z 

10.  Add  (Z>—  d)i/x,  and  (c-f  2a—  i)|/a;.  Ans.  (c-fa^/ie. 

11.  Add  (a  +  26)  m—  Cv/m,  (2a—  6c)  771—  3a^m,  (5c—  4a)  7?i— 
>/w,  and  (2a  —  3i)??i-f-4a  %/»"-.  -4?w.  (a  —  ^—  c)  ( 

12.  Add  ax-\-y-\-z,  x-\-ay-\-z,  and  x-\-y-\-az. 

Ans.  (a+2)  ( 


SUBTRACTION.  25 


SUBTRACTION.   * 

.  Subtraction,  in  Algebra,  is  the  process  of  finding  the  differ- 
ence between  two  quantities. 

53.  It  is  evident  that  5  units  of  any  kind  or  quality  subtracted 
from  8  units  of  the  same  kind  or  quality,  must  leave  3  units  of  the 
same  kind  or  quality.  That  is, 


Also,  —  8a  —  (.  —  5a)=  —  3a 

But  these  remainders  are  the  same  as  we  shall  obtain  by  changing 
the  signs  of  the  subtrahends  and  then  adding  the  results,  algebra- 
ically, to  the  minuends.     Thus, 
-f8a—  (+5a) 
—  8  a  —  (—  5a)= 
Hence,  in  Algebra, 

Subtracting  any  quantity  consists  in  adding  the  same  quantity  with 
its  sign  changed. 

54.  This  principle  may  be  established  in  a  more  general  manner 
as  follows  : 

Let  it  be  required  to  subtract  the  quantity  I  —  c  from  a. 

OPERATION.  We  first  subtract  b  from  a,  indicating  the 

Minuend,         a  operation,  and   obtain  for  a   result,   a  —  b. 

Subtrahend,       b  —  c  But  the  true  subtrahend  is  not  5,  but  b—c; 

-      and,  as  we  have  subtracted  a  quantity  too 

Difference,       a  —  b-\-c      great   by  c,   the  remainder  thus  obtained 

must  be  too  small  by  c  ;  we  therefore  add 

c  to  the  first  result,  and  obtain  the  true  remainder,  a  —  b-\-c.     But 
this  result  is  the  same  as  would  be  obtained  by  adding  —  b-\-c  to  a. 

55.  It  follows  from  the  principle  enunciated  above,  that  any 
quantity  is  subtracted  from  nothing  or  zero,  by  simply  changing  its 
sign  or  signs.     Thus, 

0—  (+a)  =  -a 
0  —  (  —  «)  —  -]-  a 
0—  (a—  b)=-a+b 

3 


26  ENTIRE    QUANTITIES. 


>.  From  these  principles  and  illustrations  we  deduce  the  fol- 
lowing 

RULE.  I.  Write  the  subtrahend  underneath  the  minuend,  placing 
the  similar  terms  together  in  the  same  column. 

II.  Conceive  the  signs  of  the  subtrahend  to  be  changed,  unite  the 
similar  terms  as  in  addition,  and  bring  down  all  the  remaining  terms 
with  their  proper  signs. 

EXAMPLES  FOR  PRACTICE. 
(1.)  (2.)  (3.)  (4.) 


6xay  —  4mca  ba*bc  Zx'y'z 

(5.)  (6.)  (7.)_ 


ax  —  2y  6a 


(8.)  (9.) 

5m—  ^a-f  c 
—  ca 


3a*x+c*d—cd*  -\-7md2 
10.  From  2xa—  3x-f  ya  subtract  a—  x*—  4x. 


11.  From  7a  —  5c-f-2  subtract  —  a-fc-f  2.  Ans.  Sa  —  6c. 

12.  From  8xa—  3xy+2/+c  subtract  x3—  6^+3.ya—  2c. 

-4ws.  7o:a+3'^—  /-f-3c. 

13.  From  a-f-6  subtract  a  —  6.  Ans.  2b. 

14.  From  ^x-\-iy  subtract  \x  —  \y.  Ans.  y. 

15.  From  a-j-6-j-c  subtract  —  a  —  b  —  c.         Ans. 

16.  From  3a-6—  2x-|-7  take  8—  36+  a-f  4x. 


17.  From  6y9—  2y—  5  take  —  8/— 

Ans.  14/-f  3y—  17. 


SUBTRACTION".  27 

18.  From  3p+q+r—  3s  take  q—  8r-f2s—  8. 

^Tis.  3p-j-9r  —  5s+8. 

19.  From  I3a*—2ax-\-9x*  take  5a3—  Tax—  x\ 

Ans.  8aa-f-5aa;-j-10a;8. 

20.  From  a4—  Sx'-j-So;9—  7x+12  take  zV-4xs-f  2xa—  6x-fl5. 

Jbis.  x3-|-3xa  —  x  —  3. 

21.  From    a6—  3a4c+5aV—  2aV-}-4ac4—  c5    take    a5-4a4c+ 
2aV—  5aV-f3ac4—  c5.  ^Lrw.  a4c4-3a3ca-|-3aV-j-ac4. 

22.  From     2x4+28x3-|-134arl—  252z-f-144    take    2x4-|-21x3+ 
67x2_63x4-84.  ^Lns.  7z3+67za—  189a;+60. 

23.  Fromx6--5x10x-lOx--54         take  x6—  5x4 


24.  From  the  sum  of  6xay  —  llax8  and  8x2^+3ax3,  take  4x>— 
«.  Ans.  10x2^  —  4ax3  —  a. 

25.  From   the   sum  of  8c<fo+15aa&  —  3    and   2cdx  —  8aa6+24 
take  the  sum  of  12aa6—  Scdx—  8  and  cdx—  4aa5+16. 

Ans.  I2cdx—  aa6  +  13. 

5T.  The  difference  of  two  dissimilar  terms  may  often  be  conven- 
iently expressed  in  a  single  term,  as  in  (51),  by  taking  some  com- 
mon letter  or  letters  as  the  unit  of  subtraction. 

EXAMPLES. 

(1.)  (2.)  (3.) 

2  ex  mx*y*  ax+by 

mx  —  4#y  ex  —  y 


(2c—  m)x  ( 

4.  From  cadam+4axa  take  tfm+Zax*. 

Ans.   (ca  — 

5.  From  ax+by+cz  take  mx+ny+pz. 

Ans.  (a  —  m)x 

6.  From  ax-\-l>x-\-cx  take  x-\-ax-\-bx.  Ans.   (c  —  l)x. 

7.  From  (a-f  26-f-c)  V~xy  take  (25—  c)Vxy. 

Ans.  (a+2c)l/xy. 

8.  From  (3a—  2m)xs-f(5a+2m)xa+(4a—  m)x  take   (a—m)x* 


28  ENTIRE    QUANTITIES. 

9.  From  l+2az«-f'3aV:f  4aV+5aV   take   z*+ 2az4+3aV-f 
4oV. 
Ans. 


USE   OF   THE  PARENTHESIS. 

SS.  The  term,  parenthesis,  will  be  employed  hereafter  as  a  gen- 
eral name  to  designate  the  various  signs  of  aggregation  employed  in 
algebraic  operations.  The  following  rules  respecting  the  use  of  the 
parenthesis  should  be  thoroughly  considered  by  the  learner,  if  he 
would  acquire  facility  in  algebraic  transformations. 

59.  From  the  definition  of  the  signs  of  aggregation,  (IT),  we 
understand  that  if  the  plus  sign  occurs  before  a  parenthesis,  all  the 
terms  inclosed  are  to  be  added,  which  does  not  require  that  the  signs 
of  the  terms  be  changed ;  but  if  the  minus  sign  occurs  before  a 
parenthesis,  all  the  terms  inclosed  are  to  be  subtracted,  which  re- 
quires that  the  signs  of  all  the  terms  be  changed.  Hence, 

1.  A  parenthesis  preceded  by  the  plus  sign  may  be  removed,  and 
the  inclosed  terms  written  with  their  proper  signs.     Thus, 

a — b  -\-  (c — d-\-  e)  =  a — b  -j-  c — d-\-  e 

2.  Conversely :  Any  number  of  terms,  with  their  proper  signs,  may 
be  inclosed  by  a  parenthesis,  and  the  plus  sign  written   before  tlie 
whole.     Thus, 

a — b-\-c — d-\-e=a-^-( — 5-j-c — d-\-e) 

3.  A  parenthesis  preceded  by  the  minus  sign  may  be  removed, 
provided  the  signs  of  all  the  inclosed  terms  be  changed.     Thus, 

a — (b — c-}-d — e)=a — b-\-c — d-\-e 

4-  Conversely :  Any  number  of  terms  may  be  inclosed  by  a  paren- 
thesis, preceded  by  the  minus  sign,  provided  the  signs  of  all  the  given 
terms  be  changed.  Thus, 

a — b  -j-  c — d-\-  e = a — b  -j-  c —  (d —  e) 

GO.  When  two  or  more  parentheses  are  used  in  the  same  express- 
ion, they  may  be  removed  successively  by  the  above  rules.     Thus, 
a — |  b — c — (d — e)  j  =a — j  b — c — d-\-e  j  =  a — b-\-c-\-d — e 

Or,  in  a  different  order, 

a — |  b — c — (d — e)  l=a — 6-f  c-\-(d — e)=a — b-\-c-\-d — e 


SUBTRACTION.  29 


EXAMPLES    FOR  PRACTICE. 

Gl.  Remove  the  parentheses  from  the  following  express-ions,  and 

reduce  the  results  : 

*: 

1.  3a-|-(2&2—  or—  d+m).  Ans.  2a+2b*—d+m. 

2.  4x»_y_(3x—  7y+5)+2x.  Ans.  4x9-f  Qy—  x—  5. 

3.  a-t-2c—  (4c—  3a-]-2m9).  ^ras.  4a—  2c—  2m9- 

4.  4x3—  2x2HX—  (2^'+5x—  7)—  6x+l]. 

Ans.  3z8+lLc—  8. 

5.  a-[-2m  —  j  c-f-x  —  [a  —  7/1  —  (c  —  2x)]  J  . 

^HS.  2a-f  m  —  2c+x. 

6.  3z9  —  4x  —  am  —  /  x8  —  a  —  [Sam  —  (2x-f  2am)-|-2a;a]  —  5am  }  . 

Ans.  4xa  —  5x-|-5a77i. 

7.  3a—  |2ma-j-[5c—  9a—  (3a+m2)]+6a—  (m2-|-5c)}. 


.  9a. 
8.  x2—  |5mca—  [a:9—  (3c—  3mc8)+3c—  (x9—  2mc3—  c)]  }. 


9.  m9  —  m  —  1  —  |  m9  —  2m  —  2—  [m9  —  3m  —  3  —  (m9  —  1m  —  4)]  }  . 

Ans.  2m-f  2. 
10.  52s—  322+4z—  1—  [2z8—  (3z9—  2z-fl)—  z*+z~\. 

Ans.  ±z*-\-z. 
11    4c»_2c9-f-c4-l—  (3c8—  c9—  c—  7)—  (c8—  4c2+2c+8). 

Ans.  3c9., 

12.  3a2t  —  4:cd  —  (Bed—  2a96)'~  [a8  +  c  —  (5crf+  3a26)  +  (3ar 
+2ca')+a3].  Ans.  8a 

13.  --   /4a9m3mV—  7m9a^—  9a9m—  TI— 


i2]—  5a9m  J—  12a9m 

Ans.  SmV+Gn—  5a9m—  3a»i9. 

.  In  Algebra,  addition  does  not  necessarily  imply  augmenta- 
tion, nor  does  subtraction  always  imply  diminution,  in  an  arithmet- 
ical sense. 

We  have  seen  that  one  quantity  is  added  to  another  by  annexing 
it  with  its  proper  sign  ;  but  a  quantity  is  subtracted  from  another 
by  annexing  it  with  its  sign  changed.     Hence, 
3* 


30  ENTIRE    QUANTITIES. 

1st  Adding  a  positive  quantity  has  the  same  effect  as  subtracting 
a  negative  quantity ;  and  adding  a  negative  quantity  has  the  same 
effect  as  subtracting  a  positive  quantity. 

2d.  If  to  any  given  quantity  a  positive  quantity  be  added,  the 
result  will  be  greater  than  the  given  quantity;  but  if  a  negative  quan- 
tity be  added,  the  result  will  be  less  than  the  given  quantity. 

3d.  If  from  any  given  quantity  a  positive  quantity  be  subtracted, 
the  result  will  be  less  than  the  given  quantity ;  but  if  a  negative 
quantity  be  subtracted,  the  result  will  be  greater  than  the  given 
quantity. 

63.  Let  — a  denote  any  negative  quantity.     Add  — b  to  this 
quantity,  and  subtract  -{-&  from  it ;  and  we  have 

—a+(-.b)=-a-b 

But  according  to  the  last  two  propositions,  the  result,  — a — 6, 
should  be  less  than  the  given  quantity,  — a.     That  is 
— a — b     < — a 

Now,  the  quantity,  — a — 5,  contains  a  greater  number  of  units 
than  — a.  These  cases,  however,  are  not  exceptions  to  the  laws 
enunciated  above ;  for  in  an  algebraic  sense,  the  less  of  two  nega- 
tive quantities  is  that  one  which  contains  the  greater  number  of 
units.  (SeelOT). 

64.  If  a  represent  the  greater  of  the  two  numbers,  and  b  the 
less,  then  a+b  is  their  sum  and  a — b  their  difference ;  and  the 
sum  and  difference  may  be  combined  in  two  ways,  as  follows  : 

1st;      To     a+b  2d;        From       a+b 

Add  a — b  Subtract  a — b 

2a  2b 

Hence, 

1.  If  the  difference  of  two  numbers  be  added  to  their  sum,  the 
result  will  be  twice  the  greater  number. 

2.  If  the  difference  of  two  numbers  be  subtracted  from  their 
mm,  the  result  will  be  twice  the  less  number. 


MULTIPLICATION.  31 


MULTIPLICATION. 

~  *' 

Go.  Multiplication,  in  Algebra,  is  the  process  of  taking  one 

quantity  as  many  times  as  there  are  units  in  another. 

G6.  In  order  to  establish  general  rules  for  multiplication,  we 
must  first  consider  the  simple  case  of  multiplying  one  monomial  by 
another;  and  we  will  investigate,  first,  The  law  of  coefficients; 
second,  The  law  of  exponents  ;  third,  The  law  of  signs. 

1st.  The  law  of  coefficients. 

Let  it  be  required  to  multiply  5a  by  36.  Since  it  is  immaterial 
in  what  order  the  factors  are  taken,  we  may  proceed  thus :  5X3=15; 
aX&— al;  and  15X«&  —  15ai.  Or  5aX3k=15a&.  Hence, 

The  coefficient  of  the  product  is  equal  to  the  product  of  the  coeffi- 
cients of  the  multiplicand  and  multiplier. 

2d.  The  law  of  exponents. 

Let  it  be  required  to  multiply  a*b*  by  cfb*.  Since  a*b*  — 
aaaa  bub,  and  a?b*=aaa  bb,  we  have 

atb3Xa3by=aaaabbbaaabb=a'Tb*.  Hence, 

The  exponent  of  any  letter  in  the  product  is  equal  to  the  sum  of 
the  exponents  of  this  letter  in  the  multiplicand  and  multiplier. 

3d.  The  law  of  signs. 

In  Arithmetic,  multiplication  is  restricted  to  the  simple  process  of 
repeating  a  number  ;  and  the  only  idea  attached  to  a  multiplier  is, 
that  it  shows  how  many  times  the  multiplicand  is  to  be  taken.  In 
Algebra,  however,  a  multiplier  may  be  affected  by  either  the  plus  or 
the  minus  sign ;  and  it  is  necessary  to  consider  how  the  sign  of  the 
multiplier  modifies  its  signification. 

For  this  purpose,  suppose  it  were  required  to  multiply  any  quantity, 
as  a,  by  c — d.  Now  it  is  evident  that  a  taken  c  minus  d  times,  is 
the  same  as  a  taken  c  times,  diminished  by  a  taken  d  times ;  or 
aX(c — d)=ac — ad.  In  the  first  term  of  this  result,  a  is  taken  c 
times  additively,  or  a-f-a+a+a  etc.,  to  c  repetitions;  and  this  is 
the  product  of  a  by  -f  c.  In  the  second  term,  a  is  taken  d  times 


32  ENTIRE   QUANTITIES. 

subtractively,  or  —  a  —  a  —  a  —  etc.,  to  d  repetitions  ;  and  this  is  the 
product  of  a  by  —  d.  Hence  we  conclude  that  the  signs,  +  an^ 
—  ,  when  prefixed  to  a  multiplier,  must  be  interpreted  as  follows  : 
The  plus  sign  before  a  multiplier  shows  that  the  multiplicand  is  to 
be  successively  added  ;  and  the  minus  sign  before  a  multiplier  shows 
that  the  multiplicand  is  to  be  successively  subtracted. 

To  exhibit  the  law  which  governs  the  sign  of  a  product,  ac- 
cording to  this  principle,  we  present  the  four  cases  which  involve  all 
the  variations  of  signs.  It  will  be  observed  that  according  to  the 
above  interpretation,  the  multiplicand  is  to  be  repeated  with  its 
proper  sign  when  the  multiplier  is  positive,  but  with  its  sign  changed 
when  the  multiplier  is  negative.  We  shall  therefore  have  the  fol- 
lowing results  : 


2.  -f-«X(  —  &)=  —  o>  —  a  —  a  —  etc.  =  —  ab. 

3.  —  aX(+&)  =  —  a  —  a  —  a  —  etc.=  —  ab. 

4.  —  «X(—  &)=+a-fa+a-fetc.  =  -fa&. 
Comparing  the  first  result  with  the  fourth,  and  the  second  with 

the  third,  we  observe  that 

When  the  two  factors  have  like  signs,  the  product  is  positive  ;  and 
when  the  two  factors  have  unlike  signs,  the  product  is  negative. 

©T.  This  law  applied  in  the  case  of  three  or  more  negative  factors 
gives  the  following  results  : 

(-a)X(-  &)  =+ab 

(_a)X(-&)X(—  c)  =(+**    )X(-O=—  o*c 

(—  «)  X  (—&)  X  (—  c)  X  (—  <0  =(—  abc  )  X  (—  *)=  +alcd 

(—a)  X  (—6)  X  (—  c)  X  (—  d)  X  (—  «)  =  (-|-a&crf)  X  (—e)=—abcde 

Hence  the  general  truth  : 

The  product  of  an  even  number  of  negative  factors  is  positive  ; 
and  the  product  of  an  odd  number  of  negative  factors  is  negative. 

CASE  I. 

68.  When  both  factors  are  monomials. 
From  the  principles  already  established  we  derive  the  following 
RULE.  I.  Multiply  the  coefficients  of  the  two  terms  together  for 
the  coefficient  of  the  product. 


MULTIPLICAT 


II.  Write  all  the  letters  of  both  terms  for 

each  an  exponent  equal  to  the  sum  of  its  exponents  in  the  two  terms. 

III.  If  the  signs  of  the  two  terms  are  alike,  prefix  the  plus  sign  to 
the  product ;  if  unlike,  prefix  the  minus  sign. 


EXAMPLES  FOR   PRACTICE. 

(1.)  (2.)  (3.)  (4.) 

lx*y  a*cm?  — 5c4m* 

§xy*          —  Gac'd  3c*d? 


5.  Multiply  17a3&V  by  lac. 

6.  Multiply  UaWc  by  10a66V. 

7.  Multiply  U7ab*c*x  by  2a86V. 

8.  Multiply  7x*yz*  by  — ±xyz.  Ans.  —2 

9.  Multiply  —  12c<f  m4  by  10c4. 

10.  Multiply  — 15a'6x8y  by  — 3a5*y. 

11.  Multiply  am  by  an. 

12.  Multiply  xmy  by  xym. 

13.  Multiply  4ambnc  by  — Ga*b*c.  Ans.  — 

14.  Multiply  Zxcyn  by  2x'cy*m.  Ans. 

15.  What  is  the  continued  product  of  3x,  2#ay,  and  7x*y3z 

Ans.  42x* 

16.  What  is  the  continued  product  of  ba'b,  ab*,  3aac,  and  — 


17.  What  is  the  continued  product  of  7a:y,  — 2xa,  3x*y,  — ory*, 
and  xy*  ? 

18.  What  is  the  continued  product  of  — 3c9e?m,  — 2cd*m,  and 
— 5cc7w3?  Ans.  —  3(W4m*. 

19.  Whac  is  the  continued  product  of  — a,  — a&,  — abc,  -~-abcd 
— abcdh,  and  — abcdhm  ? 

20.  Multiply  2(z  +y)  by  4aa(x-f  y). 

21.  Multiply  4ma<>— «)a  by  —(z—x).       Ans.  _4ma(rr— ^)8. 

22.  Multiply  (a— c)"»+1  by  (a— c)"-1.  ^ws.  (a— c)2*. 


34  ENTIRE    QUANTITIES. 

CASE   II. 

69.  When  one  or  both  of  the  factors  are  polynomials. 
1.  Multiply  x — y-\-z  by  a-\-b — c. 

OPERATION. 
X y    +Z 


Product  by    a,                         ax — ay-\-az 
Product  by    6,                                        bx — ky-{-bz 
Product  by  — c,  CX-\-Cy — CZ 


Entire  Product,  ax — ay-}-az-\-bx — by-\-bz — cx-\-cy — cz 

Hence  the  following  general 

RULE.  Multiply  all  the  terms  of  the  multiplicand  by  each  term  of 
the  multiplier,  and  add  the  partial  products. 

EXAMPLES  FOR  PRACTICE. 

a.)  (2.)  (3.) 


(4.)  (5.) 


a—  Ixy—  lOy 
6.  Multiply  3a3x2—  ifz-\-z*  by  2a^a. 


7.  Multiply  x4—  3x3-f  2x2—  5x-f3  by  3x2. 

Ans.  3a;6—  9x6+6x4— 

8.  Multiply  «V—  3a2c3+aac—  aca-f-o  —  c+1  by  or 


MULTIPLICATION.  35 

9.  Multiply  2ax—  3x  by  2x-j-4#. 

Ans.  4ax*-\-8axy  —  6xa  —  12xy. 
10.  Multiply  3a2—  2al—  6a  by  2a—  46. 


11.  Multiply  x2  —  xy+y*  by  x-f-y.  ^ns.  x8-]-?/8. 

12.  Multiply  a2  —  3ac-f-c2  by  a  —  c.      .4ns.  a3—  4a2c-f4ac2  —  c8. 

13.  Multiply  2x9—  3x-|-2  by  x—  8. 

4ns.    2x8—  19x2-f26x—  -16. 

14.  Multiply  as+2a2i+2a62+&8  by  a3—  2a26+2a&2—  1\ 

Ans.  a6  —  &fl. 

15.  Multiply  aw  +6W  by  an  -{-&". 


16.  Multiply  4x3+8x2+16x--f-32  by  3x—  6.    Ans.  12x4—  192. 

17.  Multiply  as-{-a2&-f-a&2-f  63  by  a—b.  Ans.  a4—  54. 

NOTE.  —  The  product  of  two  or  more  polynomials  may  be  indicated,  by 
inclosing  each  in  a  parenthesis,  and  writing  them  one  after  another,  with 
or  without  the  sign,  x  ,  between  the  parentheses.  Such  an  expression  is 
said  to  be  expanded,  when  the  indicated  multiplication  has  been  actually 
performed. 

18.  Expand  (a-j-m)  (a-f  d).  Ans.  cf+am+ad+dm. 

19.  Expand  (a-j-2m—  1)  (a-fl).         Ans.  aa+2am+2m—  1. 

20.  Expand  (z9  +  4s*-f-52—  24)  (z*—  4^-fll). 

Ans.  zB+151z—  264. 

21.  Expand  (a8—  4af+lla—  24)  (o"+4a-f-5). 

Ans.  a5—  41a—  120. 

22.  Expand  (m  —  3)  (m  —  1)  (m-f-1)  (w-f  3). 

Ans.  m4—  10m2  -f  9. 

23.  Expand  (x8—  2a;a-f-3a;—  4)  (4xs+3x2-j-2x+l). 


24.  Expand  (^+2^+^-4^—  11)  (y«—  2y+3). 

J.WS.  y+lOy—  33. 

25.  Expand  (c2—  c-fl)  (ca-f  c  +  1)  (c4—  c24-l). 

JTZS.  c8-f  c4-f-l. 

26.  Expand  (x6—  5x4+13x3—  x2—  x-f  2)  (x2—  2x—  2). 

Ans.  x7—  7^6-f-21x5—  17x4—  25x3-f6x3—  2x-  -4. 

27.  Expand  (16x4—  8xs-f4x2—  2x-fl)  (2x-fl). 

Ans.  32^-f-l 


36  ENTIRE    QUANTITIES. 

FORMULAS  AND  GENERAL  PRINCIPLES. 

TO.  A  Formula  is  the  algebraic  expression  of  a  general  truth 
or  principle. 

The  following  formulas  are  useful,  as  furnishing  rules  for  obtain- 
ing the  products  of  certain  binomial  factors. 

If  a  and  b  represent  any  two  quantities  whatever,  then 
a-\-b=.  their  sum,  and 
a  —  &:=  their  difference; 

and  we  have,  after  performing  the  indicated  operations,  the  results 
which  folhow: 


Or,  expressing  the  result  in  words, 

The  square  of  the  sum  of  two  quantities  is  equal  to  the  square  of 
the  first,  plus  twice  the  product  of  the  first  -and  secondj  plus  the 
square  of  the  second. 

II.  (<*—  &)f=(a—  &)  (a—  fc)  =  aa—  2a&-f-Z>9 
Or,  in  words, 

The  square  of  the  difference  of  two  quantities  is  equal  to  the 
square  of  the  first,  minus  twice  the  product  of  the  first  and  second, 
plus  the  square  of  the  second. 

III.  (a+ft)  (a—  &)=af—  V 
Or,  in  words, 

The  product  of  the  sum  and  difference  of  two  quantities  is  equal 
to  the  difference  of  their  squares. 

By  the  aid  of  these  formulas  we  are  enabled  to  write  the  square 
of  any  binomial,  or  the  product  of  the  sum  and  difference  of  any 
two  quantities,  without  formal  multiplication. 

EXAMPLES  FOR  PRACTICE. 

1.  What  is  the  square  of  3a-f-2a&? 

The  square  of  the  first  term  is  9aa,  twice  the  product  of  the  two 
terms  is  12aa&,  and  the  square  of  the  second  term  is  4a2£a  ;  hence  , 
by  the  first  formula, 

(3a+2at)a=9aa-f  12a2&H-4a959,  Arts. 


MULTIPLICATION.  37 

2.  What  is  the  square  of  2:ca—  5  ? 

The  square  of  the  first  term  is  4x4,  twice  the  product  of  the  two 
terms  is  20xa,  and  the  square  of  the  second  term  is  25  ;  hence  by 
the  second  formula, 

(2za—  5)a=4r4—  20xa+25,  An*. 

3.  What  is  the  product  of  5x-{-j/a  and  5x—  y*  't 

The  square  of  5x  is  25xa,  and  the  square  of  y*  is  #4  ;  hence  by 
the  third  formula, 

(5x+#a)  (5x-y)=25za--y,  4ns. 

4.  What  is  the  square  of  c+m?  Am.  ca-j-2cm-J-ma. 

5.  What  is  the  square  of  x  —  y  ?  Ans.  x*  —  2xy-|-<ya. 

6.  What  is  the  product  of  x-{-y  and  x  —  y  ?  Ans'.  x3  —  y*. 

7.  What  is  the  square  of  3za-f-4y  ?     Ans.  9x4-f24xV-f  ]%a. 

8.  What  is  the  square  of  5c8—  2cd  ? 

4ns.  25c6—  2 

9.  What  is  the  product  of  4^9+3^  and  42a—  Zyz  ? 


10.  What  is  the  square  of  3a*z-|-2ay  ? 


11.  What  is  the  square  of  x+l  ?  Ans.  x*  -\-2x-\-  1. 

12.  What  is  the  square  of  2^a—  1  ?  4ns.  4z4—  4za-f  1. 

13.  What  is  the  product  of  m-j-1  and  m  —  1  ?          4?is.  ma  —  1. 

14.  What  is  the  square  of  za  —  30  ?  Ans.  z*  —  60za-|-900. 

15.  What  is  the  product  of  3aa6-|-cF  and  3aa6—  <P  ? 

Ans.  9aV—  rf8 

16.  What  is  the  square  of  x  —  \y  ?  4ns.  rca  —  ccy  +  iy2. 

17.  What  is  the  square  of  2c-f-  J  ?  4ns.  4ca+2c+i. 

18.  What  is  the  square  of  xm-fy  ?  4ns.  aam+2xw/l-fty2n. 

19.  What  is  the  product  of  x"+yn  and  xm—ynt    Ans.  x^—y*". 

71.  The  binomial  square  occurs  so  frequently  in  algebraic  ope- 
rations, that  it  is  important  for  the  student  to  be  perfectly  familiar 
with  its  form.  The  higher  powers  of  any  binomial  may  be  obtained 
by  actual  multiplication.  The  3d,  4th,  and  5th  powers,  however, 
may  sometimes  be  easily  written,  without  actual  multiplication,  by 
means  of  the  formulas  which  follow  : 
4 


38  ENTIRE    QUANTITIES. 


1.  (a-f  5)i=as 

2.  (a—  6)'=a'—  3a86+3a53—  b*. 

3.  (a4-6)4= 

4.  (a—  6)4= 
5. 

6. 

Let  the  pupil  verify  the  above  by  actual  multiplication. 

73.  A  polynomial  is  said  to  be  arranged  according  to  the  descend- 
ing powers  of  any  letter,  when  the  terms  are  so  placed  that  the 
exponents  of  this  letter  diminish  from  left  to  right  throughout  all 
the  terms  that  contain  it.  Thus,  the  polynomial 

a'—  4:c4-f2z>—  x-f7 
is  arranged  according  to  the  descending  powers  of  x. 

73.  A  polynomial  is  said  to  be  arranged  according  to  the  ascend- 
ing powers  of  any  letter,  when   the  terms  are  so  placed  that  the 
exponents  of  this  letter  increase  from  left  to  right  throughout  the 
terms  that  contain  it.     Thus,  the  polynomial 

d  —  ax-\-cx*  —  bx* 
is  arranged  according  to  the  ascending  powers  of  x. 

74.  A  term  or  quantity  is  said  to  be  independent  of  any  letter, 
when  it  does  not  contain  that  letter. 

70.  The  product  of  two  polynomials  has  certain  special  proper- 
ties, which  may  be  stated  as  follows  : 

1.  —  If  both  polynomials  are  arranged  according  to  the  descending 
powers  of  the  same  letter,  then  the  first  term  obtained  in  the  partial 
products  will  contain  a  higher  power  of  this  letter  than  any  of  the 
other  terms  ;  and  as  this  term  can  not  be  reduced  with  any  of  the 
others,  it  will  form  the  first  term  of  the  entire  product. 

2.  —  If  both  polynomials  are  arranged  according  to  the  ascending 
powers  of  the  same  letter,  then  the  last  term  obtained  in  the  partial 
products  will  contain  a  higher  power  of  this  letter  than  any  of  the 
other  terms  ;  and  as  this  term  can  not  be  reduced  with  any  of  the 
others,  it  will  form  the  last  term  of  the  entire  product. 

3.  —  If  both  polynomials  are  homogeneous,  then  the  product  will 
h*.  iomogeneous  ;  and  the  degree  of  any  term  will  be  expressed  by 
the  sum  of  the  indices  denoting  the  degrees  of  its  two  factors. 


DIVISION.  39 


DIVISION. 

t; 

76.  Division,  in  Algebra,  is  the  process  of  finding  tow  many 
times  one  quantity,  called  the  divisor,  is  contained  in  another  quantity, 
called  the  dividend;  the  result  of  division  is  called  the  quotient. 

It  follows,  therefore,  that  the  quotient  must  be  a  quantity  which 
multiplied  by  the  divisor,  will  produce  the  dividend.  Thus,  revers- 
ing the  process  of  multiplication,  we  have, 

abc-^-a=bc1  because  bc^a-=abc 

7  7.  It  was  shown  in  the  multiplication  of  monomials,  (GO),  that 
the  coefficient  of  the  product  is  found  by  multiplying  together  the 
coefficients  of  the  factors  ;  and  that  the  exponent  of  any  letter  in 
the  product  is  found  by  adding  together  the  exponents  of  this  letter 
in  the  factors.  Hence,  in  division, 

1.  —  The  coefficient  of  the  quotient  must  be  found  by  dividing  the 
coefficient  of  the  dividend  by  that  of  the  divisor  ;  and 

2.  —  The  exponent  of  any  letter  in  the  quotient  must  be  found  by 
subtracting  the  exponent  of  this  letter  in  the  divisor  from  its  exponent 
in  the  dividend.     Thus, 


It  was  shown  in  multiplication,  (66),  that  when  two  factors  have 
like  signs,  their  product  is  positive  ;  and  that  when  two  factors  have 
unlike  signs,  their  product  is  negative.  In  division,  therefore,  when 
the  dividend  is  positive,  the  quotient  must  have  the  same  sign  as  the 
divisor  ;  and  when  the  dividend  is  negative,  the  quotient  must  have 
the  sign  unlike  that  of  the  divisor.  And  there  will  be  four  cases, 
with  results  as  follows  : 


2.  +ab+(—a)=—b 

3.  —  a&-K-i-a)=—  & 

4  .  —  ab  -4-  (  —  a)  =  -f-  b  Hence, 

3.  —  If  the  dividend  and  divisor  have  like  signs,  the  quotient  will 

be  positive  ;  but  if  the  dividend  and  divisor  have  unlike  signs,  the 

quotient  witt  be  negative. 


40  ENTIRE    QUANTITIES. 

'      CASE  I. 

78.  When  the  divisor  is  a  monomial. 

From  the  principles  already  given  we  have  the  following 

RULE.  To  divide  one  monomial  by  another  ;  — 

I.  Divide  the  coefficient  of  the  dividend  by   the  coefficient  »f  tin 
divisor,  for  a  new  coefficient. 

II.  To  this  result  annex  the  letters  of  the  dividend,  with  the  expo- 
nent of  each  diminished  by  the  exponent  of  the  same  letter  in  the 
divisor,  suppressing  all  letters  whose  exponents  become  zero. 

III.  If  the  signs  of  terms    are  alike,  prefix  the  plus  sign  to  the 
quotient  j  if  they  are  unlike,  prefix  the  minus  sign. 

To  divide  a  polynomial  by  a  monomial  ;  — 

Divide  each  term  of  the  dividend  separately,  and  connect  the  quo- 
tients ly  their  proper  signs. 

NOTE.  —  It  may  happen  that  the  dividend  will  not  exactly  contain  the  di- 
visor ;  in  this  case  the  division  may  be  indicated,  by  writing  the  dividend 
above  a  horizontal  line,  and  the  divisor  below,  in  the  form  of  a  fraction. 
The  result  thus  obtained  may  be  simplified,  by  suppressing  all  the  factors 
common  to  the  two  terms;  thus, 


But  as  this  process  is  essentially  a  case  of  reduction  of  fractions,  we  shall 
omit  such  examples  till  the  subject  of  fractions  is  reached. 

EXAMPLES  FOR  PRACTICE. 

1.  Divide  IQab  by  4«.  Ans.  4fc. 

2.  Divide  21aVd  by  7ac9.  Ans.  3a'J. 

3.  Divide  —  42afys4  by  6zV.  Ans.  —6x*yz. 

4.  Divide  2a°  by  a*.  Ans.  2a*. 

5.  Divide  —  a7  by  a9.  Ans.  —  a. 

6.  Divide  16^  by  kx.  Ans.  4.c2. 

7.  Divide  Ibaxy3  by  —  Say.  Ans.  —  5xy*. 

8.  Divide  117a56V  by  3a6&ca.  Ans.  39i3c. 

9.  Divide  63a8&4c<F  by  Zlabcd.  Ans.  3a'b\L 
10.  Divide  63am  by  7an.  Ans.  On"-". 


DIVISION,  4J 


11.  Divide  34:^"  by  —  17  xy.  An*.  — 

U.  Divide  (a—  c)6  by  (a—  c)8.  Ant    (a—  c)\ 

13.  Divide  35(x+y)3  by  5(z+y),  Am.  7(z+y)2. 

14.  Divide  12mV(c—  x2)5  by  3mrf(c—  a2)2. 

15.  Divide  Zlcd+lZbcx—  9&9c  by  36c.          *4w«.  cZ-f.-4.r--  35. 

16.  Divide  15a26c—  15acx2-f-5ad8c  by  —  Sac. 

17.  Divide  10x3—  15x2—  25.x  by  5.x.  An*.  2x*—  3z-5. 

18.  Divide  15x6—  45x4-f-10a;3—  105x2  by  5s;2. 

19.  Divide  amc  —  am~1c2-|-aTO-V  —  af"-V-f-af"-V  by  ac. 


20.  Divide  3m2(a— Z>)2— 3m(a— 6)  by  3(a— 6). 

Ans.  am9 — Im* — m. 

21.  Divide  7a(3m— 2a)— (3m — 2a)a  by  (3m — 2a). 

-4ns.  9a — 3m. 

CASE   II. 

79.  When  the  divisor  is  a  polynomial. 

Suppose  both  dividend  and  divisor  to  be  arranged  according  to  the 
descending  powers  of  some  letter.  Then  it  follows,  from  (75,  1), 
that  the  first  term  of  the  dividend  must  be  the  product  of  the  first 
term  of  the  divisor  by  the  first  term  of  the  quotient  similarly  ar- 
ranged. We  can  therefore  obtain  this  term  of  the  quotient,  by 
simply  dividing  the  first  term  of  the  dividend  by  the  first  term  of 
the  divisor,  thus  arranged.  The  operation  may  then  be  continued 
in  the  manner  of  long  division  in  Arithmetic;  each  remainder 
being  treated  as  a  new  dividend,  and  arranged  as  the  first. 

1.  Divide  6a4+a3Z>— 20aV+17a&8— 4Z>4  by  2a'— 3a&+ V. 

OPERATION. 

9/»». 


3a'-|-5a&— 4ia,     Quotient 


— 8aa 
-8aa 
4* 


42  ENTIRE    QUANTITIES. 

Hence  we  have  the  following 

RULE.  I.  Arrange  both  dividend  and  divisor  according  to  tht 
descending  powers  of  one  of  the  letters. 

II.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  result  in  the  quotient. 

III.  Multiply  the  wlwle  divisor  by  the  quotient  thus  found,  and 
subtract  the  product  from  the  dividend. 

IV.  Arrange  the  remainder  for  a  new  dividend,  with  which  pro- 
ceed as  before,  till  the  first  term  of  the  divisor  is  no  longer  contained, 
in  the  first  term  of  the  remainder. 

V.  Write  the  final  remainder,  if  there  be  any,  over  the  divisor  in 
the  form  of  a  fraction,  and  the  entire  result  will  be  the  quotient 
sought. 

EXAMPLES  FOR  PRACTICE. 

1.  Divide  a*+3a*x-\-3ax*+x9  by  a-f-ar.      Ans.  a*  -\-2ax-\-x*. 

2.  Divide  a3  —  4a2c-f  4ac9  —  cs  by  a  —  c.         Ans.  a9  —  3ac-fc*. 

3.  Divide  a'—  6aa-j-12a—  8  by  a3—  4a-{-4.  Ans.  a—  2. 

4.  Divide  3za—  2x4-f  #'—  x8—  2x—  15  by  x*—  5—  4x. 

Ans.  x2—  2x-f-3. 

5.  Divide  25z6—  z4—  2xf—  8*'  by  5z«—  4xa. 

Ans.  5xs+4x2-f-3x-{-2. 

6.  Divide  6a4-f9a2—  15a  by  3a2—  3a.  Ans.  2a2-f-2a-f  5. 

7.  Divide  x'—  y*  by  xs+2xV+2x/-fT^8. 

Ans.  x*  —  2x*y-\-2xy*—  y* 

3.  Divide  ax3—  (a2-f-5)x2-f69  by  ax—  b.         Ans.  x>—ax—b 
9.  Divide  a4-f464  by  a'—  2aZ>-f  2£*.  Ans.  a2-f  2ab+2b* 

10.  Divide  x6—  z4+xs—  x2-f-2x—  1  by  a2-f  x—  1. 


11.  Divide  l+3x  by  1  —  5x.    Ans.  l+8z+40x2+200x8+etc. 

12.  Divide  1—  x—  x*  by  l+x-{-z2. 

J.jw.  1—  2x-f2a;s—  2x4-\-2x* 

13.  Divide  x'—  2x»-f  1  by  x9— 


DIVISION.  43 

14.  Divide  a'-f  £'-f  c8—  3a£c  by  a+b+c. 

Ans.  a2-f-&a-r*c2  —  6c  —  ac  —  db. 

15.  Divide  2x^—  5xy~ 
by  a4—  4x3y+xy—  3.r/. 

16.  Divide  a6-fc6-fa4-f-  c4—  a4c—  ac4—  2aV  b^  a'+c3—  a2c—  ac2. 


17.  Divide  4x6—  5x*+8z4—  lOa;8—  8z2—  5x-—  4  by  4xs-f  3x2-f-2^ 
+1.  Ans.  x9—  2xs-f  3z—  4. 

18.  Divide  x*  —  a6  by  x  —  a.          J.TIS.  cc4-f-x*a-j-xa«a-j-xa8-j-a4. 

19.  Divide  a'-j-o;3  by  a  —  a. 


a  —  x 

20.  Divide  xm  —  xy^—x^y+y™  by  x—  y.     Ans.  x^—y^. 

21.  Divide  ae+m—acl"  —  amb*-\-bn+d  by  am—bn.        Ans.  ac  —  ld. 

22.  Divide  x*n—2x**y*—  2xy*n+y**  by  ccn+yn. 


EXACT  DIVISION. 

80.  Division  is  said  to  be  exact  when  the  quotient  contains  no 
fractional  part  ;  the  quotient  in  this  case  is  said  to  be  entire. 

81.  It  follows  from  the  rule  of  division,  (78),  that  the  exact 
division  of  one  monomial  by  another  will  be  impossible  under  the 
following  conditions  : 

1.  —  When  the  coefficient  of  the  divisor  is  not  exactly  contained 
in  the  coefficient  of  the  dividend. 

2.  —  When  a  literal  factor  has  a  greater  exponent  in  the  divisor 
than  in  the  dividend. 

3.  —  When  a  literal  factor  of  the  divisor  is  not  found  in  the  divi- 
dend. 

8SJ.  It  is  also  evident,  from  (79),  that  the  exact  division  of 
one  polynomial  by  another  will  be  impossible, 

1.  —  When  the  first  term  of  the  divisor  arranged  with  reference 
to  any  one  of  its  letters,  is  not  exactly  contained  in  the  first  term  of 
the  dividend  arranged  with  reference  to  the  same  letter. 

2.  —  When  a  remainder  occurs,  having  no  term  which  will  exactly 
contain  the  first  term  of  the  divisor. 


44  ENTIRE    QUANTITIES. 

GENERAL  RELATIONS  IN  DIVISION. 

8.3.  The  algebraic  value  of  a  quotient  depends  upon  the  compar« 
ative  values  and  relative  signs  of  the  dividend  and  divisor.  Now 
if  either  the  dividend  or  the  divisor  be  changed  with  respect  to  its 
value  or  sign,  the  quotient  will  undergo  a  change,  according  to  a 
certain  law.  As  these  mutual  relations  are  frequently  concerned  in 
algebraic  investigations,  we  present  them  in  this  place,  considering 
first  the  law  of  change  with  respect  to  absolute  value ;  and  second, 
the  law  of  change  with  respect  to  algebraic  signs. 

1st.  Change  of  value. 

84.  In  any  case  of  exact  division,  the  quotient  is  composed  of 
those  factors  of   the  dividend  which  are  not  included  among  the 
factors  of  the  divisor.     It  is  evident,  therefore,  that  if  we  introduce 
a  new  factor  into  the  dividend,  the  divisor  remaining  the  same,  we 
shall  introduce  the  same  factor  into  the  quotient ;  and  if  we  exclude 
a  factor  from  the  dividend,  the  divisor  remaining  the  same,  we  shall 
exclude  this  factor  from  the  quotient. 

Again,  if  we  introduce  a  factor  into  the  divisor,  we  shall  exclude 
it  from  the  quotient ;  and  if  we  exclude  a  factor  from  the  divisor, 
we  shall  introduce  it  into  the  quotient, — the  dividend  remaining  the 
same  in  both  cases. 

Hence  we  have  the  following  general  principles : 

I.  Multiplying  the  dividend  multiplies  the  quotient,  and  dividing 
the  dividend  divides  the  quotient. 

II.  Multiplying  the  divisor  divides  the  quotient,  and  dividing  the 
divisor  multiplies  the  quotient. 

III.  Multiplying  or  dividing  both  dividend  and  divisor   by  the 
same  quantity  does  not  change  the  quotient. 

2d.  Change  of  signs. 

85.  To  show  in  what  manner  the  sign  of  the  quotient  is  affected 
by  changing  the  sign  of   dividend  or  divisor,  we  observe  that  two 
signs  can  have  only  three  relations,  as  follows  : 


DIVISION.  45 

Now  if  one  of  the  signs,  only,  in  any  of  these  couplets,  be  changed, 
the  relation  of  the  signs  in  that  couplet  will  be  changed,  either  from 
like  to  unlike,  or  from  unlike  to  like ;  but  if  loth  of  the  signs  in 
any  couplet  be  changed,  their  relation  will  not  be  altered.  Hence, 

I.  Changing  the  sign  of  either  dividend  or  divisor,  changes  the 
sign  of  the  quotient. 

II.  Changing  the  signs  of  both  dividend  and  divisor,  does  not 
alter  the  sign  of  the  quotient. 


RECIPROCALS,  ZERO  POWERS,  AND  NEGATIVE  EXPONENTS 

86.  The  Reciprocal  of  a  quantity  is  the  quotient  obtained  by  di- 
viding unity  by  that  quantity.     Thus,  _    is   the   reciprocal  of  x\ 

x 

is  the  reciprocal  of  a — c. 

87.  In  dividing  any  power  of  a  quantity  by  any  other  power  of 
the  same  quantity,  we  subtract  the  exponent  of  the  divisor  from 
the  exponent  of  the  dividend,  to  obtain  the  exponent  of  the  quo- 
tient.    Thus, 


And  in  general,  we  have 

a"~~a 

If  in  this  expression  n=m,  the  exponent  of  the  quotient  will  be 
0;  and  if  n>m,  the  exponent  of  the  quotient  will  be  negathe. 
Thus, 

a3  a3  a* 

-=a'  ;    -=a'-i=a-1  ;  -=at-4=ar*l    etc. 

a  a  a 

88.  It  has  been  found  useful  for  certain  purposes  in  Algebra, 
to  employ  the  notation,  a°,  a"1,  a~2,  a~8,  etc.  We  will  therefore 
proceed  to  interpret  the  meaning  of  zero  and  negative  exponents, 
in  general. 

Let  a  represent  any  quantity,  and  m  the  exponent  of  any  power 
whatever.  Then  by  the  rule  of  division, 

£=a—  =«' 


46  ENTIRE    QUANTITIES. 

But  the  quotient  obtained  by  dividing  any  quantity  by  itself 
must  be  equal  to  1.     That  is 

<irt 

am 
Therefore,  by  Ax,  7,  we  have 

a°=l 
Hence, 

1.  Any   quantity  having  a  cipher  for  its  exponent  is  equal  to 
unity. 

Again,  by  the  rule  of  division  we  have 


But  we  have  already  shown  that  o°=l.     Substituting  this  value 
for  the  dividend,  we  obtain  the  quotient  in  another  form ;  thus, 


am     a1* 
Therefore,  by  Ax.  7,  we  have 

. — m  — 


Hence, 

2.  Any  quantity  having  a  negative  exponent  is  equal  to  the  recip- 
rocal of  that  quantity  with  an  equal  positive  exponent. 


DIVISIBILITY  OP  QUANTITIES  IN  THE  FORM  OF 

80.  There  are  certain  cases  of  exact  division  of  quantities  in  the 
form  of  am-j-Z»w  or  a1* — 6m,  which  have  important  applications 
These  may  be  exhibited  in  four  general  problems,  as  follows  : 

1.— Divide  am+lm  by  a+b. 

Commencing  the  division,  we  have 


Istrern. 


_|_  Z  =4-  ^(a^'+i1^9) 

Now  if  this  operation  be  continued,  it  is  evident  from  the  form 
which  the  first  and  second  remainders  assume,  that  when  the  expo- 


am 

+        IT 

«-f-^ 

a—  >_«»»-«& 

DIVISION.  47 

nent,  m,  is  an  odd  number,  the  mth  remainder  will  be 

—  bm(am~™—  lm~m~}=—  bm(a°—  Z>°)=—  bm(l—  1)=0 
and  the  division  will  therefore  be  exact.     But  if  m  be  even,  the  mth 
remainder  will  be 


and  the  division  will  not  be  exact.     Hence, 

The  sum  of  the  same  powers  of  two  quantities  is  divisible  by  the 
siim  of  the  quantities,  if  the  exponent  is  odd,  but  not  otherwise. 

2.—  Divide  am+bm  by  a—b. 

Commencing  the  division,  we  have 


bm 


a — b 


1st  rein.  _|_a«-'&+          b1*  =-f&(a'*-1-f  61"-1 

--a"*"1^  —  a"^~*b 


2drem. 


If  this  operation  be  continued,  it  is  evident  that  whether  m  be 
odd  or  even,  the  mth  remainder  will  be 


and  the  division  can  not,  therefore,  be  exact.     Hence, 

The  sum  of  the  same  powers  of  two  quantities  is  never  divisible 

by  the  difference  of  the  quantities. 
3. — Divicje  am — bm  by  a-{-b. 
Commencing  the  division,  we  have 

a+b 


1st  rem.  aM-'&  —          lm  = b(a1*'1-\-bm~1') 

i^b 


2drem.  ,      _f-a«-V—  bm  ==-±.b*(a'»-'>—bm-*) 

If  this  operation  be  continued,  then  it  is  evident  that  when  m  is 
odd,  the  mth  remainder  will  be 

— 7/»(ar»-m-f-&m-nl)  —  —  6m(a°+60) =—  &m(l-f-l)=— 2bm 
and  the  division  can  not  be  exact.     But  if  m  be  even,  the  mth  re- 
mainder will  be 

_|_j«(tt«— _j«-«)  =  _|_&»(a«_&»)=_|_j*(l__l)  =  0 

and  the  division  in  this  case  will  be  exact.     Hence, 


48 


ENTIRE    QUANTITIES. 


The  difference  of  the  same  powers  of  two  quantities  is  divisible  by 
the  sum  of  the  quantities,  if  the  exponent  is  even,  but  not  otherwise. 
4.  —  Divide  am  —  bm  by  a  —  b. 
Commencing  the  division,  we  have 


—       6« 


am       —  < 


a—b 


1st  rem. 


2drem.  _[_»«*-»&»_          fr»  —_}_&'  (a»*-»_&*-') 

If  this  operation  be  continued,  then  it  is  evident  that  whether  m 
be  odd  or  even,  the  rath  remainder  will  be 

-f  lm(cr-™—  I™-™}  =  -j-bTO(a°—  Z>°)  =  -f  6m(l—  1)  =0 
and  the  division  will  therefore  be  exact.     Hence, 

The  difference  of  the  same  powers  of  two  quantities  is  always 
divisible  by  the  difference  of  the  quantities. 

OO.  If  we  continue  the  division  in  the  1st,  3d,  and  4th  of  the 
preceding  problems,  then  in  the  cases  of  exact  division,  the  form 
of  the  quotients  will  be  as  follows  : 


a  —  b 


(1) 


«—  b™-1   (2) 
*-i+&"~1    (3) 


91.  By  giving  particular  values  to  m  in  (1),   (2)  and  (3),  we 
•btain  the  following  results,  which  maybe  useful  for  reference: 


a-f  6 


(2) 


DIVISION. 


a—b 


=a*+a1>+l* 


In  like  manner  we  may  obtain 


(3) 


?— 1  .        , 
=x — 1 


x+l 

ft          "1 

£=T 
i " 

!h=i: 

a:*— 1 
a;  — 1 


(5) 


(6) 


FACTORING. 

The  Factors  of   a  quantity  arc  those   quantities  which, 
being  multiplied  together,  will  produce  the  given  quantity. 

OS.  A  Prime  Factor  is  one  which  can  not  be  producedjby  the 
multiplication  of  two  or  more  factors;  it  is  therefore  divisible  only 
by  itself  and  unity. 

5  D 


50  ENTIB.E    QUANTITIES. 


.  An   algebraic  expression  may  be  factored  by  inspection, 
by  trial,  or  by  its  law  of  formation. 

To  express  the  prime  factors  of  a  monomial,  we  have  only  to 
factor  the  coefficient,  and  repeat  each  letter  as  many  times  as  there 
are  units  in  its  exponent.  Thus, 

15a8x*y  =3  X  5  X  aaaxxy 

@5.  The  following  remarks  will  aid  in  factoring  polynomials  : 
Is*.  If  all  the  terms  of  a  polynomial  have  a  common  factor,  the 
quantity  may  be  factored  by  writing  the  other  factors  of  each  term 
within  a  parenthesis,  and  the  common  factor  without.     Thus,    • 

2«V—  6aV-f4a2.T—  10a'=2aa  (x8—  3xa+2x—  5a) 
2d.  If  two  of  the  terms  of  a  trinomial  are  perfect  squares,  and 
the  other  term  is  twice  the  product  of  the  square  roots  of  the 
squares,  the  trinomial  will  be  the  square  of  the  sum  or  difference  of 
these  roots,  (7O,  I  and  II),  and  may  be  factored  accordingly.  Thus, 
in  the  trinomial,  4a4  —  20o2&-{-2562,  the  two  terms,  4a4  and  25&a, 
are  the  squares  of  2a2  and  5&  respectively,  and  the  other  term, 
20a26  is  equal  to  2x2aaX5&;  and  we  have 

4a4—  20a2i-h2562^(2aa—  5&)(2a8—  56) 

3d.  If  a  binomial  consists  of  two  squares  connected  by  the  minus 
sign,  it  must  be  equal  to  the  product  of  the  sum  and  difference  of 
the  square  roots  of  the  terms,  (7O?  III).  Thus, 

9x2—  y»=(3x+y)  (3.r—  y) 

4th.  Quantities  in  the  form  of  aw±&w  may  be  factored  by  refer- 
ence to  the  principles  and  formulas  relating  to  these  quantities. 

Thus, 

as+Z,8=(a-f&)  (a2—  a&+  Z>2) 

NOTE.  —  It  may  happen  that  when  there  is  no  factor  common  to  all  the 
terms,  a  portion  of  the  polynomial'may  be  factored. 

EXAMPLES    FOR    PRACTICE. 

1.  Factor  a'6-f  a262-f-a26c.  Ans.  a'i(a 

2.  Factor  3xy~  3xy+3;ey—  6^y. 

Ans. 

3.  Factor  5a86c2—  15aa6V—  5aaicV. 

Ans. 


DIVISION.  51 

4.  Factor  a'-j-c'tf-f  cmx.  Ans.  a'+c(c+m)x. 

5.  Factor  x*—  x'y+xy'—y*.  /^Vy^-v/ 

6.  Factor  a4&a-j-2a86'-}-aa&4.  Ans.  '  a'l*(a-\-b)  (a+fi). 

7.  Arrange  (x9  —  x')a-\-(x*-\-x)(5b—c)—q  according  to  the  pow- 
ers of  x.  Ans.  (a-f-3&  —  c)ajf—  ^a  —  36+c)ic  —  #. 

8.  Factor  a'm  —  9awi8.  Ans.  am(a*  —  3m)  (aa-{-3m). 

9.  Factor  8a8—  x8.  4™.  (4a"+2cKC+xs)  (2a—  »). 

10.  Factor  y'-}-243.      ^TIS.  (y«—  Sy'+O.y1—  27y+8l)  (y+3). 

11.  Find  the  factors  of  x'—  y*. 

Ans.  (a^+ay+y)  (*«—  ay+y")  (a;+y)  (x—  y). 

12.  Find  the  factors  of  a8—  a&*-f-2a&c—  ac*. 

J.TIS.  a(a-|-&—  c)  (a  —  &+c). 

SUBSTITUTION. 

00.  Substitution,  in  Algebra,  is  the  process  of  putting  ono 
quantity  for  another,  in  any  given  expression. 

1.  Substitute  y—  1  for  x,  in  x'-\-x*—  5z  —  3. 

OPERATION. 


—  5x  =—  5(y—  1)  = 

=  --3 


=/—  2^—4^+2,     Ans. 

ITence,  for  substitution  we  have  the  following 

RULE.  Perform  the  same  operations  upon  the  substituted  quantify 
as  the  expression  requires  to  be  performed  upon  the  quantity  for 
which  the  substitution  is  made 

EXAMPLES  FOR  PRACTICE. 

1.  Substitute  a  —  b  for  a  in  aa-fa&-[-3*.          Ans.  a?  —  a5-f  b*. 

2.  Substitute  a-f  2  for  a  in  a*—  2a-fl.  Ans.  ccs-(-2x-f  1. 
8.  Substitute  a-f  3  for  y  in  y4—  2y8+y"~6. 

.  .T4-f  lOa'- 


52  ENTIRE    QUANTITIES. 

4.  Substitute  s+r  for  x,  in  xa-j-az-|-&,  and  arrange  the  result  ac- 
cording to  the  descending  powers  of  r. 

Ans.  ra-f(2s+a>-f  sa-j-as-}-£. 

5.  What  will  a4-f  a8&+a76a+a&'+&4  become,  when  I—  a  ? 

Ans.  5a*. 

6.  What  will  x*-f  oxa-f  a*x-{-a*  become,  when  m-f-1  is  put  for  x 
and  m  —  1  for  a  ?  Ans.  4m(raa+l). 

7.  What  will  x4-f-y4  become,  when  a-}-b  is  put  for  x  and  a  —  b 
fory?  ^ns.  2(a4+6a2&'+64). 

8.  What  is  the  value  of  (x+0+b+c)*+  (z—  a—  6—  c)6,  when 


9.  In  x8  —  7a+6  substitute  y—  2  for  a.       -4ns.  y'—  6y'-|-5y. 
10.  In  x6—  2*4-f  3z»—  7x'+8a;—  3   substitute  y+1  for  a;. 


11.  If  a  —  6=z,  b  —  c=y,  and  c—  a  =  as,  prove  that  2(o  —  6)* 


THE  GKEATEST  COMMON  DIVISOR 

97.  A  Common  Divisor  of  two  or  more  quantities  is  a  quantity 
which  will  exactly  divide  each  of  them. 

98.  The  Greatest  Common  Divisor  of  two  or  more  quantities 
is  the  greatest  quantity  that  will  exactly  divide  each  of  them }  it  is 
composed  of  all  the  common  prime  factors  of  the  quantities. 

The  term,  greatest,  in  this  connection,  is  used  in  a  qualified  sense, 
and  has  reference  to  the  degree  of  a  quantity,  or  of  its  leading  term, 
not  to  its  algebraic  or  its  arithmetical  value.  Thus,  if  x — 3  and 
iC3_|_4x-j-2  are 'the  prime  factors  common  to  two  or  more  quantities, 
then  according  to  the  above  definition,  (xa+4o;-j-2)(a; — 3)=#8-{- 
xa — 10x — 6,  is  the  greatest  common  divisor.  But  this  product  is 
not  necessarily  greater  in  value  than  one  of  the  prime  factors.  For, 
if  a;— 4,  then  we  have 

»»-f 4x-f2=34,  and  x8-j-o;a— lOz— 6=34. 

99.  Several  quantities  are  said  to  be  prime  to  each  other  when 
they  have  no  common  factor. 


DIVISION.  53 

CASE  I. 

1OO.  "When  the  given  quantities  can  be  factored  by 
inspection. 

It  is  evident  from  (81.  2)  that  no  factor  of  the  greatest  common 
divisor  can  have  an  exponent  greater  than  the  least  with  which  it 
enters  the  given  quantities.  Hence  the  following  obvious 

EULE  I.  Find  ~by  inspection,  or  otherwise,  all  the  different  prime 
factors  that  are  common  to  the  given  quantities,  and  affect  each  with 
the  least  exponent  which  it  has  in  any  of  the  quantities. 

II.  Multiply  together  the  factors  thus  obtained,  and  the  product 
will  be  the  greatest  common  divisor  required. 

EXAMPLES  FOR  PRACTICE. 

1.  Find  the  greatest  common  divisor  of  a* — 2a*x*-}-ax'>  and  a4— 
2aax-\-a*x*. 

Factoring,  we  have 

a6— 2aV+a  x4=a  (a4— 2aV+  z4)==a(a— x)\a+ x)9 
a4— 2a8x  -f-aV=aa(aa— 2ax  -f  xa)=aa(a— xf 

The  lowest  powers  of  the  common  factors  are  a  and  (a — x)9 ;  and 

we  have 

a(a — x)2 = a8 — 2aax + axa 

the  greatest  common  divisor  required. 

2.  Find   the   greatest   common   divisor   of  2a*bc*,    6a&V,    and 
IQa3bc\  Ans.  2abc\ 

3.  Find   the   greatest  common   divisor   of  5rE2,y V,    6x8^2,  and 
I2x*yz*.  Ans.  x*yz*. 

4.  Find  the  greatest  common  divisor  of  x* — y3  and  xa — 2xy-\-y* 

Ans.  x — y. 

5.  What    is    the   greatest   common   divisor   of  a2m — b*m   and 
2ac2m — 2c*bm  ?  Ans.  m(a — &). 

6.  What  is  the  greatest  common  divisor  of  aax8 — 3aaxa+a2a:  and 

az3  ?  Ans.  a(x2— 3z+l). 

7.  What  is  the  greatest  common  divisor  of  16xa — 1,  x — 4xa,  and 

?  Ans. 


54  ENTIRE   QUANTITIES. 

CASE    II. 

101.  "When  the  given  quantities  can  not  be  factored  by 

inspection. 

102.  The  greatest  common  divisor  is  found  in  this  case  by  a 
process  of  decomposing  the  quantities  by  division.     But  in  order  to 
deduce  a  rule  for  the  method,  it  will  be  necessary  first  to  establish 
certain  principles  relating  to  exact  division. 

103.  First,  suppose  A  to  be  a  quantity  which  is  exactly  divisible 
by  another  quantity,  D,  and  let  q  represent  the  quotient.     Then, 

A 

D=q 
If  we  now  multiply  the  dividend  by  m,  we  shall  have,  from  (84:  I), 

^=gm 
D      " 

in  which  qm  is  entire.     Thus  we  have  shown  that  if  D  divides  A, 

it  will  also  divide  Am.     Hence, 

l.I/a  quantity  will  exactly  divide  one  of  two  quantities,  it  will 
divide  their  product. 

Again,  let  A  and  B  represent  any  two  quantities,  and  S  their  sum. 
Now  suppose  both  A  and  B  are  exactly  divisible  by  D,  and  let 

-=2,  and  yy=2'-     We  shall  have 
And  dividing  each  term  by 


Of 

in  which  _  must  be  entire,  because  its  equal,  q-{-qr,  is  entire.  Hence, 

2.  If  a  quantity  will  exactly  divide  each  of  two  quantities,  it  will 
divide  their  sum. 

Finally,  let  d  represent  the  difference  of  A  and  B,  and  suppose 
A  and  B  to  be  divisible  by  D,  q  and  qe  being  the  quotients,  as 

before.     We  shall  have 

A—  B=d 

And  dividing  every  term  by  D, 


in  which  _  is  entire,  because  q  —  q'  is  entire.     Hence, 


DIVISION.  55 

3.  If  a  quantity  icill  exactly  divide  each  of  two  quantities,  it  will 
divide  their  difference. 

1O4.  We  may  now  show,  by  the  aid  of  these  principles,  wln-t 
relation  the  greatest  common  divisor  of  two  quantities  bears  to  the 
parts  of  these  quantities  when  decomposed  by  Division. 

Suppose  two  polynomials  to  be  arranged  according  to  the  powers 
of  the  same  letter,  and  let  A  represent  the  greater  and  B  the  less. 
Then  let  us  divide  the  greater  by  the  less,  the  last  divisor  by  the 
last  remainder,  and  so  on,  till  nothing  remains.  If  we  represent  the 
several  quotients  by  q,  q',  q",  etc.;  and  the  remainders  by  R,  R',  R"j 
etc.,  the  successive  operations  will  appear  as  follows  : 

(1.)  (2.)  (3.) 

By**        *)«  *'» 


R  R?  0 

To  investigate  the  mutual  relations  of  A,  B,  Ry  and  R'j  we  ob- 
serve that  in  division  the  product  of  the  divisor  and  quotient,  plus 
the  remainder,  if  any,  is  always  equal  to  the  dividend.  Hence,  from 
the  operations  above,  we  have  the  three  following  conditions  : 

R'q"        =R 

Rq' 

Bq 

Now  from  the  first  equation  it  is  evident  that  R'  divides  R  with- 
out remainder;  it  will  therefore  divide  Rq',  (1O3?  1).  And  since  Rr 
divides  both  Rq'  and  itself,  it  must  divide  their  sum,  Rq'-\-R',  or 
B,  (1O3,  2)  ;  consequently,  it  will  divide  Bq,  (1O3,  1).  Finally, 
since  it  divides  both  Bq  and  R,  it  must  divide  their  sum,  jBq-\-JR, 
or  A,  (1O3,  2).  Hence, 

I.  The  last  divisor,  R',  is  a  common  divisor  of  R,  B}  and  A,  or 
of  all  the  dividends. 

Again,  the  dividend  minus  the  product  of  the  divisor  and  quo- 
tient, is  always  equal  to  the  remainder.     Therefore,  from  the  first 
and  second  operations  above,  we  have 
A—Bq  =R 


56  ENTIRE   QUANTITIES. 

Now  any  expression  which  will  divide  B,  will  divide  By,  (1O3,  1)  ; 

hence,  any  expression  which  will  divide  both  A  and  B,  will  also  di- 
vide A  —  Bq,  or  R,  (  1O8,  3).  Whence  it  follows  that  the  greatest 
common  divisor  of  A  and  B  will  divide  R,  and  is  therefore  a  common 
divisor  of  B  and  R.  For  like  reasons,  referring  to  the  second  equa- 
tion, the  greatest  common  divisor  of  B  and  R  will  also  divide  R',  and 
is  therefore  a  common  divisor  of  R  and  R'.  But  the  greatest  common 
divisor  of  R  and  R'  is  Rf  itself.  Consequently,  R'  is  the  greatest 
common  divisor  of  R  and  B}  and  also  of  B  and  A.  Hence 

II.  The  last  divisor,  Rf,  is  the  greatest  common  divisor  of  the 
given  quantities,  and  also  of  the  dividend  and  divisor  in  each  subse- 
quent operation. 

1.  What  is  the  greatest  common  divisor  of  12x8  —  2x*  —  7-r  —  3 


FIRST   OPERATION. 


12zs—  2x2—  Ix—  3 
12x*—2x*—4x 


Go; 


2— 2x— 


X 1     1st  Rem. 

SECOND  OPERATION. 
'— 2x— 1  X— 1 


Ans.  x — 1. 

The  process  here  employed  for  finding  the  greatest  common  divi- 
sor of  two  polynomials,  is  subject  to  two  modifications,  which  we  will 
now  investigate  in  their  order. 

1st.  Suppressing  monomial  factors. 

It  is  evident  that  any  monomial  factor  common  to  the  given  poly- 
nomials, may  be  suppressed  in  both,  and  set  aside  as  one  factor  of 
their  greatest  common  divisor.  We  may  then  apply  the  process  of 
division  to  the  resulting  polynomials,  and  obtain  the  remaining  fac- 
tor or  factors  of  the  greatest  common  divisor  required. 

Again,  if  either  polynomial  contains  a  factor  which  is  not  common 
to  both,  this  factor  can  form  no  part  of  the  greatest  common  divisor 


DIVISION. 


57 


required,  and  may  therefore  be  suppressed.  And  since  the  greatest 
common  divisor  of  the  given  polynomials  is  the  same  as  that  of  the 
dividend  and  divisor  in  each  operation  following  the  first,  (II),  it 
is  evident  that  we  may  suppress  the  monomial  factors  in  every 
remainder  that  occurs.  And  it  should  be  observed,  that  if  all  the 
monomial  factors  of  the  given  quantities  have  been  previously  sup- 
pressed, no  monomial  factor  of  any  one  of  the  remainders  can  belong 
to  the  greatest  common  divisor  sought,  or  be  common  to  any  two  suc- 
cessive remainders,  (II).  This  modification  of  the  process  will  be 
illustrated  by  the  example  which  follows  : 

2.  What  is  the  greatest  common  divisor  of  12x6-f-22x3-f-6x  and 
6xB— 15xs— 36x? 

The  first  polynomial  contains  the  monomial  factor  2x,  and  the 
second  contains  the  monomial  factor  3x.  We  therefore  suppress 
these  factors,  setting  aside  x,  which  is  common,  as  one  factor  of  the 
greatest  common  divisor  sought.  We  then  apply  the  process  of  di- 
vision to  the  resulting  polynomials,  as  follows : 

FIRST  OPERATION. 


6x4-fllx9-L  3 
6x4— 15x'— 36 


2x4— 5xa— 12 


26xa+39 

Suppressing  the  factor  13  in  this  remainder,  we  have  2xa-f-3  for 
the  next  divisor. 

SECOND  .OPERATION. 

a— 12        2xa+3 


2x4-f3xa 


— 8xa— 12 
— 8xa— 12 


Taking  the  last  divisor,  and  the  common  factor,  x,  which  was  set 
aside  at  the  beginning,  we  have 

(2xa-f-3)Xa=2x'-|-3x.  Ans. 

2d.  Introducing  monomial  factors. 

It  may  happen  at  any  stage  of  the  process,  that  after  suppressing 
every  monomial  factor  of  the  divisor,  its  first  term  will  not  be 
exactly  contained  in  the  first  term  of  the  dividend.  In  such  cases, 
the  dividend  may  be  multiplied  by  such  a  factor  as  will  render  its 
first  term  divisible  by  the  first  term  of  the  divisor.  No  factor  thus 


58  ENTIRE    QUANTITIES. 

introduced  can  be  common  to  the  dividend  and  divisor,  since  by 
hypothesis  all  the  monomial  factors  of  the  divisor  have  previously 
been  suppressed.  Consequently,  if  the  process  of  division  be  con- 
tinued under  this  modification,  the  last  divisor  must  be  the  greatest 
common  divisor  sought.  This  point  will  be  illustrated  by  the  fol- 
lowing example  : 

3.  What  is  the  greatest  common  divisor  of  2x* — 12x3-f-17x2-}- 
Qx— 9  and  4z8— 18x2-fl9z—  3  ? 

"We  first  multiply  the  greater  polynomial  by  2,  to  render  its  first 
term  divisible  by  the  first  term  of  the  other  polynomial. 


FIRST     OPERATION. 


4— 24x8-f34za+12:r— 18 


— 3 


,-1 


—  2z3-f-  5z2-t-  5x—  6 

4-10x— 12      New  prepared  dividend. 

a— 19a-  3 


In  the  above  operation,  we  suppress  the  facto.r  3  in  the  first 
remainder,  and  multiply  the  result  by  2,  to  render  the  first  term 
divisible  by  the  first  term  of  the  divisor.  We  thus  obtain  — 4x3-f- 
10x24-10x  — 12  for  the  second  dividend.  As  the  two  partial 
quotients,  x  and  — 1,  have  no  connection,  they  are  separated  by  a 
comma. 

Multiplying  the  last  divisor  by  2  for  a  new  dividend,  we  proceed 
as  follows : 

SECOND    OPERATION. 

— 8xa+29x— 15 


s—  36x2+  38x-     6 


—  7x2+  23x—    6 

—  56x2-}-184x 48        New  prepared  dividend. 

-105 


—  I9x+  57 

Dividing  this  remainder  by  — 19,  We  have  x — 3  for  the  next 
divisor. 


DIVISION.  59 

THIRD    OPERATION. 

X 3 


15 
15 


Thus  we  find  that  the  greatest  common  divisor  is  x — 3.  Had 
we  suppressed  -f-19  instead  of — 19,  in  the  final  remainder  of  the 
second  operation,  we  should  have  obtained  — .z-f-3,  or  3 — x  for  the 
greatest  common  divisor.  It  should  be  remembered,  however,  that 
the  term  greatest,  in  this  connection,  has  reference  to  exponents 
v  and  coefficients,  and  not  to  the  algebraic  value ;  (O8).  Consequent- 
ly either  x — 3,  or  3 — x  may  be  considered  the  greatest  common 
divisor  of  the  given  polynomials.  And  it  is  immaterial  what  sign 
is  given  to  any  monomial  factor  which  we  may  suppress  or  introduce 
at  any  stage  of  the  work. 

1O*>.  From  these  principles  and  illustrations  we  deduce  the  fol- 
lowing general 

RULE.  I.  Arrange  the  two  polynomials  with  reference  to  the  same 
letter  ;  then  suppress  all  the  monomial  factors  of  eacht  and  if  any 
factor  suppressed  is  common  to  the  two  polynomials,  set  it  aside  as 
one  factor  of  the  common  divisor  sought. 

II.  Divide  the  greater  of  the  resulting  polynomials  by  the  less, 
and  continue  the  division  till  tlie  first  term  of  the  remainder  is  of  a 
lower  degree  than  the  first  term  of  the  divisor  ;  observing  to  suppress 
the  monomial  factors  in  every  remainder,  and  to  introduce  into  any 
dividend,  if  necessary,   such  a  factor  as  will  render  its  first  term 
exactly  divisible  by  the  first  term  of  the  divisor. 

III.  Take  the  final  remainder  in  the  first  operation  as  a  new 
divisor,  and  the  former  divisor  as  a  new  dividend,  and  proceed  as 
before  ;  and  thus  continue  till  the  division  is  exact.      The  last  divisor, 
multiplied  by  the  common  factor,  if  any,  set  aside  at  the  beginning, 
will  be  the  greatest  common  divisor  required. 

TV.  If  more  than  two  polynomials  are  given,  find  the  greatest 
common  divisor  of  the  first  and  second,  and  then  the  greatest  com- 
mon divisor  of  this  result  and  the  third  polynomial,  and  so  on. 
The  last  will  be  the  greatest  common  divisor  required. 


60  ENTIRE   QUANTITIES. 


EXAMPLES  FOR  PRACTICE. 

Find  the  greatest  common  divisor, 

1.  Of  x4— 2x3— 4*2-f  llx— 6  and  x3— 8x9-fl7x— 10. 

Ans.  x*— 3x42. 

2.  Of  6x3-|-x2— 44x+21  and  6x3— 26x'-f  46x— 42. 

4n*.  3x— 7. 
8.  Of  x8— 6ax2-f-10aax— 3a3  and  3axa— 14aax  -J-15a«. 

J.jis.  x — 3a. 

4.  Of  x4— 8xs-f  14x2-fl6x— 40  and  x9— 8xa-j-19x— 14. 

5.  Of  a3-j-5a2+5a+l  and  as-fl.  Ans.  a-fl. 

6.  Of  2a4— 5a3i— 3a^3-f7ai8  +  3i4and4a3— 2aa5— 4a&a— 3 

^4?is.  2a— 3i. 

7.  Of  3x8— 4xV+3^2— 2?/3  and  4x2— 7^+3.ya. 

Ans.  x — y. 

8.  Of  4x6— 2x4-f-4x3— 27x9+4o;— 7  and 
—5.  Ans. 

9.  Of  a*c — 4a3cm-f  3acm3  and  aV— 6aacam-|-5c2m9. 

^4»s.  c(aa — m). 

10.  Of  x4— 4xs— 16xa+7x+24  and  2x3— 15xa-h9x-j-40. 

Ans.  xa— 5x — 8. 

11.  Of  15x94-71o;4-f  60xa— 56  and  3x6— 17x4— 20za+84. 

Ans.  3x2-j-7. 

12.  Of  3a44-14a2ma— 5wi4,  6a4— 14a3ma-f4»i4,  and  3a4— 22a2m« 
-f-7m*. 

13.  Of  2aV— 2 

^u?w.  ax — by. 

14.  Of  9a4-f  12a8+10a3+4a-f-l  and  3a4-f-8a3-}-14a2+8a-f-3. 

Ans. 


I 


LEAST  COMMON  MULTIPLE. 

1O6.  A  Multiple  of  any  quantity  is  another  quantity  exactly 
divisible  by  the  given  quantity. 

It  follows  from  this  definition  that  if  one  quantity  is  a  multiple 
of  another,  the  multiple  must  be  equal  to  the  product  of  the  other 


DIVISION.  61 

quantity  by  some  entire  factor.     Thus,  if  A  is  a  multiple  of  JB,  then 
A=JBm)  in  which  m  is  entire. 

107.  A  Common  Multiple  of  two  or  more  quantities  is  one 
which  is  exactly  divisible  by  each  of  them. 

108.  The  Least  Common  Multiple  of  two  or  more  quantities 
is  the  least  quantity  which  is  exactly  divisible  by  each  of  them. 

CASE  I. 

109.  "When  the  quantities  can  be  factored  by  inspec- 
tion. 

From  the  principles  of  exact  division,  we  may  make  the  follow- 
ing inferences  : 

1.  —  A  multiple  of  any  quantity  must  contain  all  the  factors  of  that 
quantity. 

2.  —  A  common  multiple  of  two  or  more  quantities  must  contain  all 
the  factors  of  each  of  the  quantities. 

3»—  The  least  common  multiple  of  two  or  more  quantities  must 
contain  all  the  factors  of  each  of  the  quantities,  and  no  other  factors. 

Hence  the  following 

RULE.  I.  Find  by  inspection  all  the  different  prime  factors  that 
enter  into  the  given  quantities,  and  affect  each,  with  an  exponent  equal 
to  the  greatest  which  it  has  in  any  of  the  quantities. 

II.  Multiply  together  the  factors  thus  obtained,  and  the  product 
will  be  the  least  common  multiple  required. 

EXAMPLES   FOR  PRACTICE. 

1.  What  is  the  least  common  multiple  of  »*-}-«&,  a?d  —  b*d,  and 


Factoring,  we  have 

a3   -f  ab  —a(a-\-b) 

a?d—b*d  =<?a—  6 


The  highest  powers  of  the  different  prime  factors  are  a,  d,  c, 
(a  —  &)',  and  (a-j-6)  ;  and  we  have 

(a-f  6)=a4c<2—  a86cd—  ctb'cd+ab'cd,  Ans. 


62  ENTIRE    QUANTITIES. 

2.  Find  the  least  common  multiple  of  2a46c,  5aV,  lOczfc'rf,  and 


3.  Find  the  least  common  multiple  of  3#2y,  15xy2,  IQxyz3,  and 


4.  Find  the  least  common  multiple  of  x*-\-xy,  xy  —  y  ,  and  x*  —  y*  . 

Ans.  x*y  —  xi/9. 

5.  Find  the  least  common  multiple   of    x*  —  a4,  x2  —  a2,  x2-|-«a, 
and  a;4—  2aV+a*.  Ans.  x8—  a2x4—  aV-j-a6. 

6.  Find  the  least  common  multiple  of  x*  —  x,  x8  —  1,  and  re'-f-l.   I4-  " 

7.  Find  the  least  common  multiple  of  x4+2xa-|-l,  x4  —  2x2-{-l, 
^_j_2x-f-l,  x1—  2x-f-l,  «-f  1,  and  x—  1.  ^ln*.  x8—  2x4-j-l. 

8.  What  is  the  least  common    multiple   of  4ar*4-2x,  6xa  —  4x, 
and'6x2+4a;?  J.ws.  36x'-f  2x3—  8x. 

9.  ,\Vhat  is   the   least  common  multiple  of  x2  —  4a9,  (x-j-2a)s, 
and  (x—  2a)«  ?  X6-  '  ^^^^  %  ^  -  l>  k<f 

10.  What  is  the  least  common  multiple  of  a*  —  &*,  a*  —  5s,  a*  —  68, 
and 


CASE   II. 

11O.  "When  the  quantities  can  not  be  factored  by 
inspection. 

The  rule  for  this  case  may  be  deduced  as  follows : 

1. — If  two  polynomials  are  prime  to  each  other,  their  product 
must  be  their  least  common  multiple. 

2- — If  two  polynomials  have  a  common  divisor,  their  product 
must  contain  the  second  power  of  this  common  divisor ;  their  least 
common  multiple  will  therefore  be  obtained,  by  suppressing  the  first 
power  of  the  common  divisor  in  the  product,  or  in  one  of  the  given 
quantities  before  multiplication. 

3. — If  w.e  find  the  least  common  multiple  of  two  polynomials,  and 
then  the  least  common  multiple  of  this  result  and  a  third  polyno- 
mial, and  so  on,  the  last  result  will  evidently  contain  all  the  factors 
of  the  given  polynomials,  and  no  other  factors.  It  will,  therefore, 
be  the  least  common  multiple  of  the  polynomials  (1OO,  3). 


DIVISION.  63 

Hence  the  following 

RULE.  I.  When  only  two  polynomials  are  given  : — 

Find  the  greatest  common  divisor  of  the  given  polynomials; 

suppress  this  divisor  in  one  of  the  polynomials,  and  multiply  the 

result  by  the  other  polynomial. 

II.  When  th'ree  or  more  polynomials  are  given  : — 
Find  the  least  common  multiple  of  any  two  of  the  polynomials  ; 
then  find  the  least  common  multiple  of  this  result  and  a  third  poly- 
nomial j  and  so  on,  till  all  the  polynomials  have  been  used.      The 
last  result  will  be  the  least  common  multiple  required. 

NOTE. — It  will  generally  be  found  preferable  to  commence  with  the 
greatest  and  next  greatest  of  the  given  quantities. 


EXAMPLES   FOR  PRACTICE. 

Find  the  least  common  multiple 

1.  Of  x'-fx9— 4x+6  and  x'— 5x'-f8x— 6. 

Ans.  x4— 2x8— 7x9-f-18x— 18. 

2.  Of  x8— 2x9— 19x-}-20  and  x9— 12x-f35. 

Ans.  x4— 9x8— 5x9-fl53x— 140. 

3.  Of  6a'm4— am9— 1  and  2a'm4-f3am8— 2. 

Ans.  6a'w6-f-llafm4—3am»— 2. 

4.  Of  2x'— 5x'— x+l  and  x8-5xa-|-7x— 2. 

Ans.  2x4— 9*s+ 9x94-3x— 2. 

5.  Of  3z'+6x'— 5x— 10  and  6x4— 4x2— 10. 

Am.  6x6+12x4— 4x3— 8xa— lOx— 20. 

6.  Of  x'-f  7x+10,  xa— 2x— 8,  and  xa-f  x— 20. 

Ans.  x3-f-3x9— 18x— 40. 

7.  Of  a9— 3a6-f-251,  a9— ab— 26s,  and  a9—  b\ 

Am.  a*—2a*b—a  b* + 26s. 

8.  Of  2x»— 7xy-f-3y9,  2x'— 5xy-|-2#9,  and 

Ans.  2x8— 


64  FRACTIONS. 


FRACTIONS. 

DEFINITIONS  AND  NOTATION. 

111.  We  have  seen    (12)  that  division  may  be   indicated  by 
writing  the  dividend  and  divisor  on  opposite  sides  of  a  horizontal 
line.     The  term  Fraction,  in  Algebra,  relates  to  this  mode  or  form 
of  indicating  division.     Hence, 

112.  A  Fraction  is  a  quotient  expressed  by  writing  the  dividend 

above  a  horizontal  line,  and  the  divisor  below.     Thus  —  is  a  frac- 

b 
don,  and  is  read,  a  divided  by  b. 

113.  The  Denominator  of  the  fraction  is  the  quantity  below  the 
line,  or  the  divisor. 

114.  The  Numerator  is  the  quantity  above  the  line,  or  the 
dividend. 

».  Any  fraction  may  be  decomposed  as  follows : 


Hence, 

1.  —  The  value  of  a  fraction  is  equal  to   the  r-eciprocal  of  the 
denominator  multiplied  by  the  numerator. 

2.  —  In  any  fraction,  the  reciprocal  of  the  denominator  may  be 
regarded  as  a  fractional  unit  ;  and  the  numerator  shows  how  many 
times  this  unit  is  taken  in  the  fraction.     Hence, 

3.  —  A  fraction  is  a  fractional  unit  or  a  collection  of  fractional 
units,  the  value  of  each  depending  upon  the  denominator. 

11O.  An  Entire  Quantity  is  an  algebraic  expression  which  has 
no  fractional  part  ;  as  x2  —  3xy. 

.  A  Mixed  Quantity  is  one  which  has  both  entire  and  frac- 


tional parts;  as  a*  _j_  —. 


GENERAL   PRINCIPLES.  65 


GENERAL  PRINCIPLES  OF  FRACTIONS. 

118.  Since  a  fraction  is  a  form  of  expressing  division,  it  is  evi- 
dent that  all  the  operations  in  fractions  mu§fc  be  based  upon  the 
general  relations  subsisting  between  the  dividend,  divisor,  and 
quotient.  These  principles  relate,  first,  to  change  of  value  ;  second, 
to  change  of  sign. 

1st.  Change  of  value. 

110.  By  modifying  the  language  of  (84),  we  may  express  the 
mutual  relations  of  the  numerator  and  denominator  of  a  fraction,  as 
follows  : 

I.  Multiplying  the  numerator  multiplies  the  fraction,  and  dividing 
the  numerator  divides  the  fraction. 

II.  Multiplying  the  denominator  divides  ike  fraction,  and  dividing 
the  denominator  multiplies  the  fraction. 

111.  Multiplying  or  dividing  both  numerator  and  denominator  by 
the  same  quantity,  does  not  alter  the  value  of  the  fraction. 

2d.  Change  of  sign. 

12O.  The  Apparent  Sign  of  a  fraction  is  the  sign  written 
before  the  dividing  line,  to  indicate  whether  the  fraction  is  to  be 
added  or  subtracted.  Thus,  in  » 


the  apparent  sign  of  the  fraction  is  plus,  and  indicates  that  tho 
fraction  is  to  be  added. 


.  The  Real  Sign  of  a  fraction  is  the  sign  of  its  numerical 
value,  when  reduced  to  a  monomial,  and  shows  whether  the  fraction 
is  essentially  a  positive  or  a  negative  quantity.  Thus,  in  the  fraction 
just  given,  let  a=2>  and  x=.  3.  Then 


4a— 2x       8^-6  2 

The  real  sign  of   this  fraction  therefore   is   minus,  though  its 
apparent  sign  is  plus. 

132.  Each  term  in  the  numerator  and  denominator  of  a  fraction 
has  its  own  particular  sign,  distinct  from  the  real  or  apparent  sign 
6*  E 


66  FRACTIONS. 

of  the  fraction.     Now  the  essential  sign  of  any  entire  quantity  is 
changed,  by  changing  the  signs  of  all  its  terms.     Hence, 

I.  Changing  all  the  signs  of  either  numerator  or  denominator , 
changes  the  real  sign  of  ike  fraction;  (8«f?  I). 

II.  Changing  all  the  signs  of  both  numerator  and  denominator, 
does  not  alter  the  real  sign  of  the  fraction;  (8o?  II). 

III.  Clianging  the  apparent  sign  of  the  fraction,  changes  the  real 
sign. 


REDUCTION. 

123.  The  Reduction  of  a  fraction  is  the  operation  of  changing 
its  form  without  altering  its  value. 

CASE   I. 

124.  To  reduce  a  fraction  to  its  lowest  terms. 

A  fraction  is  in  its  lowest  terms,  when  the  numerator  and  denom- 
inator are  prime  to  each  other.  And  since  it  does  not  alter  the 
value  of  a  fraction  to  suppress  the  same  factor  in  both  numerator 
and  denominator,  (HOj  III),  we  have  the  following 

RULE.  I.  Resolve  the  numerator  and  denominator  into  their 
prime  factors,  and  cancel  all  those  factors  which  are  common 
Or, 

II.  Divide  both  numerator  and  denominator  by  their  greatest 
common  divisor. 

EXAMPLES   FOR  PRACTICE. 

1.  Reduce  to  its  lowest  terms 


_- 

*    ' 


a'-j-a3"      a'(a2-|-l)  a 

2.  Reduce        ^7"^"^  —  ^   to  its  lowest  terms. 
4a*  —  2a*  —  3a-f  1 


KEDT7CTION.  67 

The  greatest  common  divisor  of  the  numerator  and  denominator, 
as  found  by  (1O5),  is  a — 1 ;  hence, 

(3aa— 2a— l)-4-(a— l)=3a+l 
(4tt»_2aa— 3o+l)  -T-  (a— 1)= 4aa-f  2a— 1 

And  we  have  for  the  reduced  fraction, 

3a+l 
— a — • j    Ans. 

Reduce  each  of  the  following  fractions  to  its  lowest  terms  : 


3.    ^.  Ans.   £ 


4     X*~~l.  Ans.   *—1 

*y+y  y 

a* — al*  A        a* — ab 

5.    — .  jSLns. 


x6— &V  A 

6.   — r — j-  •  Ans. 

2X* — I6x — 6 

7 ^Ins.-  _. 

'•    3*'— 24x— 9  3 


«  Ans 

°-    4X» — 4x»_f     •  ••=* 


v<          s    i    o     2;     10      / a    i     ;  » 

a-j-o 


10. 


a8— xa  a-fx 

11.    ^^ 


x'-^-.+l^  ^ns.^1.. 

"•    a;«_x'_x2-fa;  x 


•    (x-fy)s-x«-y  3 


(3xa— 1)  (2xa— 1)— xa(5x>— 7)  .  1 

U-: 1 1 .; L+  Ans.    

•  XO        9  t-\9      l       /       •  O        N»  •      I 


OS  FRACTIONS. 

CASE  H. 

125.  To  reduce  a  fraction  to  an  entire  or  mixed  quan- 
tity. 

The  division  indicated  by  a  fraction  may  be  at  least  partially 
performed,  when  there  is  any  term  in  the  numerator  whose  literal 
part  is  exactly  divisible  by  some  term  in  the  denominator.  Hence, 

RULE  I.  Divide  the  numerator  by  the  denominator  as  far  as  pos- 
sible, and  the  quotient  will  be  the  entire  quantity. 

II.  Write  the  remainder  over  the  denominator,  annex  the  fraction 
thus  formed  to  the  entire  part,  with  its  proper  sign,  and  the  whole 
result  will  be  the  mixed  quantity. 

EXAMPLES   FOR  PRACTICE. 

Reduce  the  following  fractions  to  entire  or  mixed  quantities  : 

x 

Ans.  a-f  T-. 

bx 

Ans.  a-\ • 

a 

Ans.  5a+l+— . 
y 

Ans.  2( 


Ans.  x—  4a—  3-f-^ 
ox 


-    Sx'—  7x'+7z+30      <-H~  3x-10 

•  An>-  3*5 


REDUCTION.  69 

10.  ^  ' 

*—y 

x«_6.r4-f  10*'— 3 


CASE  m. 

To  reduce  a  mixed  quantity  to  the  form  of  a 
fraction. 

This  case  is  the  converse  of  the  last,  and  may  be  explained  by  it. 

Hence  the  following 

RULE.  Multiply  the  entire  part  by  the  denominator  of  the  frac- 
tion ;  add  the  numerator  if  the  sign  of  the  fraction  be  plus,  but 
subtract  it  if  the  sign  be  minus,  and  write  the  result  over  the  denomi- 
nator. 

EXAMPLES   FOR  PRACTICE. 

Reduce  the  following  mixed  quantities  to  fractions : 


2.  2b-^=^  Am. 

c 


b  o 


An,. 


6.  3o — 9 — 

a+3  a+3 

— ^ — .  Ans.  . 

x—y  x—y 


(x— 2)'  a— 2 


TO  FRACTIONS. 

9.  a>aJ>l>-°L±.  An,.  - 


o—6 

2y4-4-2y§  1+y 

1 A      "I     I    Q,.    1    Q  -.9    I    f>j.8    I        *y !____.  ^i?lS. 


CASE   IV. 


.  To  transfer  a  factor  from  the  denominator  to  the 
numerator,  or  the  reverse. 


Let  us  take  any  fraction,  as  -  ,  and  multiply  both  numerator 

by* 

and  denominator  by  ^~n,  observing  that  any  factor  having  zero  for 
its  exponent  is  equal  to  unity,  (88?  1),  and  may  therefore  be  omitted. 
We  shall  have 


ax 


If  we  multiply  both  numerator  and  denominator  of  the  same  frao 
tion  by  x~mj  we  shall  have 

axm     axm~m        ax*  a 


In  like  manner  we  may  transfer  any  factor  having  a  negative  ex 

ponent.     For  example,  let  us  take  the  fraction,  ax    ,  and  multiply 

b 

both  numerator  and  denominator  by  x'  ;  we  shall  have 


b       ~  bx°        bx'~bx' 

By  the  same  principle  also,  any  fraction  may  be  reduced  to  the 
form  of  an  entire  quantity  ;  thus, 


In  all   operations  of  this  kind,  the  intermediate  steps  may  be 
omitted,  and  the  results  obtained  by  the  following 


REDUCTION.     *-.  71 


RULE.  I.  To  transfer  any  factor  from  the  denominator  to  the  nu- 
merator, or  the  reverse  : —  Change  the  sign  of  its  exponent. 

II.  To  reduce  any  fraction  to  the  form  of  an  entire  quantity : — 
Transfer  all  the  factors  of  the  denominator  to  the  numerator,  ob- 
serving to  change  the  signs  of  the  exponents  of  the /actors  transferred. 


EXAMPLES    FOR    PRACTICE. 

In  each  of  the  following  fractions,  transfer  the  unknown  factors, 
or  factors  containing  unknown  quantities,  to  the  numerator. 

ax  axi/~* 

* 


c 


3a9  .        3a3*-' 

Ans. 


1. 


5ni 

3  _I_  AI  *-*-'-. 

axyz  a 

•  c 


4A  . 

Ans. 


> 
5cm(a  —  ar)~*  5c 


4m 

In  each  of  the  following  fractions,  transfer  the  known  factors  to 
the  denominator,  and  the  unknown  factors  to  the  numerator. 

«'&cV 
o     -  ,  Ans. 


(x — a)  l(a — ^)"  (a — &)* 

HAns.  -     1— ^ 
•  •              7 i      ft    A                                                                                                                          r i     o» ft 


72  FRACTIONS. 

In  the  following  fractions,  transfer  the  factors  having  negative 
exponents. 

12.  »£5i  Ans.    *?L 

5m*-« 

13.  M*'-l) 

' 


14 

*  '* 


Reduce  each  of  the  following  fractions  to  the  form  of  an  entire 
quantity. 

15.        !1  An*.  5a 


16.  L^L.  Ans. 
am* 

17.  *L. 


18. 


CASE  V. 

128.  To  reduce  one  or  more  fractions  to  a  common 
denominator. 

We  have  seen  (124)  that  a  fraction  may  be  reduced  to  lower 
terms  by  division.  Conversely,  a  fraction  must  be  reduced  to  higher 
terms  by  multiplication,  and  each  of  the  higher  denominators  it  may 
have,  must  be  some  multiple  of  its  lowest  denominator.  Hence, 

1. — A  common  denominator  to  which  two  or  more  fractions  may 
be  reduced,  must  be  a  common  multiple  of  their  lowest  denomina- 
tors ;  and 

2. — The  hast  common  denominator  of  two  or  more  fractions, 
must  be  the  least  common  multiple  of  their  denominators. 

1.  Reduce  _f_  and  — .to  their  least  common  denominator. 
a*b         al* 


REDUCTION.  73 

"We  find  by  inspection  that  the  least  common  multiple  of  the 
given  denominators  is  a*ba.     And 


If,  therefore,  we  multiply  both  numerator  a^d  denominator  of  the 
first  fraction  by  I9,  and  of  the  second  by  a,  we  shall  reduce  the  two 
fractions  to  their  least  common  denominator,  cfl?.     Thus, 
cXb*=b*c,  new  numerator  of  first  fraction  ; 
:  new  numerator  of  second  fraction. 


TT  c        d          b*c       ad       A 

Hence  _  ,    ___  .    _  ,  Ans. 

a*b     aV       a*b*     a*b* 

From  these  principles  and  illustrations  we  deduce  the  following 
RULE.  I.  find  the  least  common  multiple  of  all  the  denominators, 

for  tlie  least  common  denominator. 
II.  Divide  this  common  denominator  by  each  of  the  given  denom- 

inators, and  multiply  each  numerator  by  the  corresponding  quotient. 

The  products  will  be  the  neic  numerators. 

NOTE.  —  Mixed  numbers  should  first  be  reduced  to  fractions,  and  all 
fractions  to  their  lowest  terms. 


EXAMPLES  FOR  PRACTICE. 

In  each   of  the  following  examples,  reduce   the  fractions  and 
mixed  quantities  to  their  least  common  denominator : 

2a  36  4ac     Bbx 

1.  —    and    — ~  Ans.  —,    -=—• 

x  2c  2cx     2cx 


4ac 

Ans.  — , 
2c 


5«     3& 


4.    «    ^±1,  and 


b' 

acx  H-  aac       (bx  -\-b}  (x  -j-  a)  icy 

'  &c.r  -j-  a6c  bcx  ~\-  abc  bcx -j-  abc 


T 


CJL 

y  ay — 1  ay' — y        ay' — y 


FRACTIONS. 


—  5         x1  —  4x'  —  4*  —  5 


x—  5 


ADDITION. 

12O.  We  have  seen  (115)  that  a  fraction  is  equal  to  the  re- 
ciprocal of  its  denominator  multiplied  by  the  numerator.  Hence, 
if  two  or  more  fractions  have  a  common  denominator,  they  will  have 
a  common  fractional  unit,  which  may  be  made  the  unit  of  addition. 
Thus, 

a        b       1  1  1 


Or  thus, 


a        b  a-\-b 

-  +  -  =  ac-'-f  bcrl=(a+V)crl=  - 


The  intermediate  steps  may  be  omitted  ;   hence  the  following 

RULE.  I.  Reduce  the   fractions  to  their  least    common  denomi- 
nator. 

II.  Add  the  numerators,  and  -write  the  result  over  the  common 
denominator. 

NOTES.  1.  If  there  are  mixed  quantities,  we  may  add  the  entire  and 
fractional  parts  separately. 

2.  Any  fractional  result  should  be  reduced  to  its  lowest  terms. 


ADDITION.  75 


EXAMPLES    FOR    PRACTICE. 

3*     2x         * 
LAdd-g,    _and_.  An*. 

_    a      ,a+b  ac+ab+b* 

2.  Add  Tand—  -  —  Am.  -  - 

be  vc 

a1      .  a*+a? 

3.  Add  -z  and  —  f  —  Ant. 

a+x 


. 
4.  Add  —     and 


a-b"^a+b  '    aa-6a 

i— 5 


5.  Add  2cH — £-and 

*— 2  .  2x— 3         M       „     .  5x*— 4x— 9 

6.  Add  5x+-^— and 


7.  Add  JL,    J£-,and-^-.  An*.  ±tf. 

a-f-c     a  —  c         a-j-c  a  —  c 

8.  Add 


9*  Add  (b  —  c)    (c  —  a)'    (c  —  a)  (a  —  £)'  &nd  (a—  4)  (i—  c)' 

Ans.  0. 

Aril  a'~  b  &'+«  ,          1-f  a6 

'  'ai 


.  0. 


be  ac  , 

11.  Add   ,  -  —  -  -  -,     -  -  T-T-,  -  r,and 


12.  Add 


(a  —  b)  (a  —  c)     (6  —  c)(i  —  a)'        (c  —  a)(c  — , 

X 3  05  —  2  03—1 

P         xa- 

J.71S.    — -^ 


— 6 

;8— 2-a— 3x 


76  FRACTIONS. 


SUBTRACTION. 

13O.  If  two  fractions  have  a  common  denominator,  they  will 
have  the  same  fractional  unit ;  and  the  one  may  be  subtracted  from 
the  other,  by  taking  the  difference  of  the  numerators.  Thus, 


c         c       c  c  c 

Or  thus, 

a— b 


Hence  the  following 

RULE.  I.  Reduce  the  fractions  to  their  hast  common  denominator 
II.   Subtract  the  numerator  of  the  subtrahend  from  the  numerator 
of  the  minuend,  and  write  the  result  over  the  common  denominator. 


EXAMPLES  FOR  PRACTICE. 

3x              x  2z  13* 

i     From    — -  subtract  — -.  -Ans.  -^-. 

7              y  v)o 

7X                2z— 1  17^4-2 

2.    From  —  subtract  — - —  Arts. -! — . 

23  o 

0     From  subtract  — ; — .  Ans.  — j-. 


Ua—  10    .  .    '       ,    3a—  5 

4.  From  3a  -\  --  —  —    take  2a  -)  --  -  —  . 

lO  7 

a  4-  b     ,     a  —  6 

5.  1<  rom  -  -  take  -  -—  _  •  An*,  -r  -  — 

0_  l          a  _|_  5  a*  —  b* 

6.  From    x  +    *~y    take  %±JL.       Ant.  x  —  -J?- 

*  *  '*— 


-,         0      .    x  ,  .  x  —  a 

7.    From  3x  -J  —  take  ic  --  . 
be 

x9  +  x  —  5  a;*  4-  x  —  1 

1 


-llx  +  12 

4x+  8 


9.  From 


MULTIPLICATION.  77 

3a-f-6 


,  2(a  —  5) 

J.NS. 


.A    T, 

10.  From  . 


7a6(a— 6)  —  2(a'  —  4') 


MULTIPLICATION. 

131.  Any  fraction  may  be  multiplied  by  an  entire  quantity  in 
two  ways  : 

1st.  By  multiplying  its  numerator;  or 

2d.  By  dividing  its  denominator  ;  (119,  I  and  IT). 

139.  A  general  rule  for  the  multiplication  of  fractions  is  fur- 
nished  by  the  following  example  : 

1.  Multiply  ~  by  1. 
6  a 

OPERATION. 


6      d 

By  observing  the  result,  we  find  that  the  new  numerator  is  the 
product  of  the  given  numerators,  and  the  new  denominator  is  the 
product  of  the  given  denominators.  Hence  the  following 

RULE.  I.  Reduce  entire  and  mixed  quantities  to  fractional  forms. 

II.  Multiply  the  numerators  together  for  a  new  numerator,  and 
the  denominators  for  a  new  denominator,  canceling  all  factors  com- 
mon to  the  numerator  and  denominator  of  the  indicated  product. 

EXAMPLES  FOR  PRACTICE. 

_    ,,  ...  ,      a   ,      b  a 

1.  Multiply   -  by  —  Ans.  —• 

b          x  x 


78  FRACTIONS. 


0    _,  .  .  .          -         ,     2a 

3.  Multiply   -  by  _.  Ans. 

4.  Multiply   ^L^r'by-^--  A*. 

2y  a-fx 

5.  Multiply  L_ 


<•    nr  ,..  ,     o—     ,          a 

6.  Mulbply    _by  ___  Ans. 


1.  Multiply         =by  2"  ^~-  8(-+«). 

8.  Multiply  a+X-  by  a--. 


II  I* 

9.  Multiply   *X'-?X    by    — — Ans.   *ax~*>a  t 

*  J  1J.  "V        <?^»       Q~  4X* g 


10.  Multiply  together    —  —V-  ,   —  ^-,    and  —  ^  —       Ans.  a. 
x        x+y  x—y 


..Multiply   l^J^lby  — _f6 


Sa' 

55 


?* 


12.  Multiply  together   -±,   ±= 
Za       a-j-6 


Ans. 


13.  Multiply  together    - — ^-,   -^ .and  a-| — ^-. 

a  -|-6      ax-\-x*  a — x 


x 

14.  Multiply  together    ^- — ^ ,    a    *"    .  and  <^H_ . 

a* — 6a     a' -fa;*          a — x 

4»t. 

a»_Z,»    aa-|-6*        ,    *- 

15.  Multiply  together    — = — ,        '     ,  and 


, 

UC  O--C  X  -  O 


J+c 


DIVISION. 
16 


.  Multiply  together      ,c(a~C)    t>       *"+*>   ,,  and  £=1*. 
rj  aa-j-2uc-fc*     a'—  2ac-j-ca  ac2^ 




ax 


Ar  ii-   i       "-' — (;rt — -c    u 
.  Multiply  I  T        A         -r     by 


& — c  (c — &•— a)(6 — c — a) 


DIVISION. 

.  Any  fraction  may  be  divided  by  an  entire  quantity  in  two 
ways  : 

1st.  By  dividing  its  numerator  ;  or 

2d.  By  multiplying  its  denominator;   (119,  I  and  II). 

We  may,  however,  derive  a  general  rule  for  the  division  of  frac- 
tions, from  the  following  example  : 

1.  Divide  —  by  —  • 
6          d 

OPERATION. 

•4=*^.-rf4g=g 

6      d  cd~l      Ic 

By  inspecting  this  result,  we  find  that  the  new  numerator  may  be 
obtained,  by  multiplying  the  numerator  of  the  dividend  by  the  de- 
nominator of  the  divisor  ;  and  the  new  denominator  may  be  obtained, 
by  multiplying  the  denominator  of  the  dividend  by  the  numerator 
of  the  divisor.  Hence  the 

RULE.  I.  Reduce  entire  and  mixed  quantities  to  fractional  forms. 

II.  Invert  the  terms  of  the  divisor,  and  proceed  as  in  multiplica- 
tion. 

EXAMPLES  FOR  PRACTICE. 

.    _.  ..     5x  ,      I  .       §xc        5cx 

1.  Divide  —  by  —  Ans.  —  Xr  =  —  r 


a      o          ao 


by  -1-,. 

c        *   a-f  b 


80  FRACTIONS. 


. 

8.  Divide  -  by  -r  -  ;-  An*-  —  -7-^ 

*       *  2c 


a  —  x         a—  x 


2xa— 7  ,               a*  (2xa— 7)  (x+a) 

4.  Divide  — ; —  by  nrns — ;— •'    Ans-    « 

x+a      J   xa-j-2ax-f-aa  a* 

5.  Divide     3  X'~l\  78  by  ^±  ^?is.  xa+Z>a. 

x* — 2bx-4-l*     J   x — b 


2ax+ x*  ,         x 
6.  Divide  -H^-   by  - 


70x— 15 
Ans.  -TT- -• 

lO.r — 4 

8.  Divide  ~~  .         by  Jr  -4.w*.  - 

6x— 7        x— 1  18av-21 

9.  Divide  — — y  by  — —  Ans.  — - — - — . 

^-  -j    '«+»*   ,      2ax-f2axa  7 

10.  Divide  -g^-  by  -    —  ^Lns.   ^ 

11.  Divide  -5-^ 


12.  Divide  -^-^  by  t- 

0  0 

wo —  TIX         ma — mx 

13.  Divide  -  — -7-  by 


a-j-6  *m 

x  a 


1*     -n-    -j       3(x7— 1)  /X+l\  (X—I\ 

15.  Divide  -7-7 — —-  by   I  -^r1—  1 1 7  ]  -  Ans.  3a. 

2(a-f-i)      J    \  2a  /  \a-\-b/ 


IP    iv  M     10^-f3a'+3^a 
16.  Divide 


,„    _.  .,     aa      1    ,       a      1       1 

17.  Dmde  -3-f-  by  -,--+-•  ^, 


18.  Divido 

a 


4ns.  1. 


COMPLEX  FORMS.  81 


REDUCTION   OF   COMPLEX   FORMS. 

134.  A  fraction  is  said  to  be  simple,  when  both  numerator  and 
denominator  are  entire ;  otherwise  it  is  said  to*  be  complex. 

1  Ho.  To  reduce  a  complex  to  a  simple  fraction,  we  may  regard 
the  quantity  above  the  line  as  a  dividend,  and  the  quantity  below  it 
as  a  divisor,  and  proceed  according  to  the  last  rule. 

A  more  convenient  method  may  be  derived  from  the  following 
observations : 

1. — If  a  fraction  be  multiplied  by  its  own  denominator,  the 
product  will  be  the  numerator. 

2. — If  a  fraction  be  multiplied  by  any  multiple  of  its  denomina- 
tor, the  product  must  be  entire. 

Hence  to  simplify  a  complex  fraction,  we  have  the  following 
RULE.  Multiply  both  numerator  and  denominator  by  the  least 
common  multiple  of  the  denominators  of  the  fractional  parts. 


EXAMPLES  FOR  PRACTICE. 


1.   Simplify 


a-j-a? 

Multiplying  both  numerator  and  denominator  by  (a — a;) 
or  by  its  equal  a* — x*,  we  have 


a      _i 

a  —  x  a*4-ax  —  a*4-x*     ax-\-x* 

'  ' 

.. 


- 

—  .  __!  —  ,    Ans. 


a         a?  —  xa  —  a*-\-ax     ax  —  x9      a  —  x 


2.   Simplify  __i.  Ans. 


82  FRACTIONS. 


ac 


3.   Simplify  ±_JL'.  ^s_ 


i'c  +  ao3 


4.  Simplify 


m  n 


n 


a  —  6 


+ 


+ 


0+1    (    X—l  <rra-U»^-L(w_m) 


5.   Simplify 

~~  a—b—i 


5.   Simplify  if gc        q6  .      A 

V    *    ab        ac        be  An*' 


7.   Simplify   ^ ?=?.  .       «c—i^ 

«+&   ,  «~^  Ans'  -ac+Td' 


S.   Simplify   - I lil-  a5 

a+6         a— 6  ^w*. 

a — b         a+6 
1 


9.   Simplify   X~L      X-T\  A       x(y*- 

1  1 


a+1      6+1      c+1     c?+l 
10.    Simplify    _? L___f 1. 


SIMPLE   EQUATIONS.  83 


SECTION  II. 


SIMPLE  EQUATIONS. 

13O.  An  Equation  is  an  expression  of  equality  between  two 
quantities.     Thus, 

x+y=a 
is  an  equation,  signifying  that  the  sum  of  x  and  y  is  equal  to  a. 

137.  The  First  Member  of  an  equation  is  the  quantity  on  the 
left  of  the  sign  of  equality  ;  and 

The  Second  Member  is  the  quantity  on  the* right  of  the  sign  of 
equality.     Thus,  in  the  equation, 

x—  3y=a— b, 

the  first  member  is  x — 3y,  and  the  second  member  is  a — b.     The 
two  members  are  sometimes  called  the  two  sides  of  the  equation. 

138.  It  is  important  to  observe  that  the  kind  of  equality  sub- 
sisting  in  an    equation  is  algebraic;  that  is,  the  two  members  must 
have  the  same  essential  sign,  as  well  as  the  same  arithmetical  value. 

139.  The  Unknown  Quantity  of  an  equation  is  the  letter  to 
which  some  particular  value  or  values  must  be  given,  in  order  that 
the  statement  contained  in  the  equation  may  be  true.     And  such 
value  or  values  are  said  to  satisfy  the  equation.     An  equation  may 
contain  two  or  more  unknown  quantities. 

1 4O.  ,A  Boot  of  an  equation  is  any  value  which,  being  substi- 
tuted for  the  unknown  quantity,  will  satisfy  the  equation.     For  ex- 
ample, in 

z2+2z=35 

let  us  substitute  5  for  x ;  we  shall  then  have 
5'+  2X5=35 
25  +10=35 
35=35 


84  SIMPLE    EQUATIONS. 

Hence,  5  is  a  root  of  the  equation,  because  if  substituted  for  #,  it 
will  render  the  two  members  equal.     Again,  let  x=.  —  7.     We  have 
(-7)'+(-7x2)=35 
49—14=35 
35=35 
Hence,  —  7  is  also  a  root  of  the  given  equation. 

141.  A  Numerical  Equation  is  one  in  which  all  the  known 
quantities  are  expressed  by  figirres  as,  3x3  —  x'-\-2x=17. 

142.  A  Literal  Equation  is  one  in  which  some  or  .  all  of  the 
known  quantities  are  expressed  by  letters  ;  as  ax'  —  36x=5rZ. 

143.  An  Equation  of  Conditi&n  is  one  which  must  exist  be- 
tween certain  known  or  arbitrary  quantities,  in  order  that  certain 
other  equations  may  be  true.     Thus,  the  two  equations, 

x-\-c=ba 
•  x  —  c=  a 

can  not  both  be  true  at  the  same  time,  unless 

c=2a 

That  is,  the  last  equation  expresses  the  condition  which  will  render 
the  other  two  equations  true  ;  it  is  therefore  called  an  equation  of 
condition. 

144.  An  Identical  Equation  is  one  in  which  the  two  members 
are  the  same  algebraic  expression,  or  are  reducible  to  the  same. 
Thus, 


a8—  aa=(z-f-a)  (x—  a) 
are  identical  equations. 

145.  Equations  are  said  to  be  of  different  degrees  or  dimen- 
sions. 

The  Degree  of  an  equation  is  denoted  by  the  greatest  number  of 
unknown  factors  occurring  in  any  term.  Hence, 

1.  —  If  an  equation  involves  but  one  unknown  quantity,  its  degree 
is  denoted  by  the  highest  exponent  of  this  quantity  in  any  term. 

2.  —  If  an  equation  involves  more  than  one  unknown  quantity,  its 
degree  is  denoted  by  the  greatest  sum  which  the  exponents  of  the 
unknown  quantities  give  in  any  term. 


TRANSFORMATION.  85 

Thus,  for  example ; 

x-\-ax  —  b       ") 

S-  arc  equations  of  the  first  degree ; 
ax+y  =  c*      3 

=  8      •) 

are  equations  of  the  second  degree; 


are  equations  of  the  third  degree. 

146.  A  Simple  Equation  is%an  equation  of  the  first  degree. 

147.  A   Quadratic  Equation  is  an  equation  of  the  second 
degree. 

148.  A  Cubic  Equation  is  an  equation  of  the  third  degree. 


TRANSFORMATION  OP  EQUATIONS. 

149.  The  Transformation  of  an  equation  is  the  process  of 
changing  its  form  without  destroying  the  equality  of  its  members 

From  the  nature  of  an  equation,  it  is  evident  that  all  the  opera- 
tions to  which  it  can  be  subjected  without  destroying  the  equality, 
are  embraced  in  the  axioms  (39)  ;  they  may  be  stated  as  follows: 

1. — The  same  or  equal  quantities  may  be  added  to  both  members ; 
(Ax.  1). 

2. — The  same  or  equal  quantities  may  be  subtracted  from  both 
members.;  (Ax.  2). 

3. — Both  members  may  be  multiplied  by  the  same  or  equal 
quantities ;  (Ax.  3). 

4 — Both  members  may  be  divided  by  the  same  or  equal  quanti- 
ties ;  (Ax.  4). 

5. — Both  members  may  be  raised,  by  involution,  to  the  same  pow- 
eJr;  (Ax.  8). 

6. — Both  members  may  be  reduced,  by  evolution,  to  the  same 
root ;  (Ax.  9). 
8 


86  SIMPLE    EQUATIONS. 


CASE   I. 

15O.  To  transpose  the  terms  of  an  equation. 

Transposition  is  the  process  of  changing  a  term  from  one  member 
of  an  equation  to  the  other,  without  destroying  the  equality. 

To  exhibit  the  law  of   transposition,  let  us  consider  the  three 
following  examples : 

1. — Let  x-\-a  =  b. 

If  we  subtract  a  from  both  members  of  this  equation,  the  result 
will  be 

x  =*b — a; 

and  we  perceive  that  the  term,  -j-a,  has  been  removed  from  the  first 
member,  and  appears  as  — a  in  the  second  member. 

?.— Let  x— a  =6. 

If  we  add  a  to  both  members  of  this  equation,  the  result  will  be 

x  =  b-\-a } 

and  we  perceive  that  the  term,  —a,  has  been  removed  from  the  first 
member,  and  appears  as  -f~a  iQ  the  second. 

3. — Let  a — x  =  b. 

Subtracting  a  from  both  members  of  the  equation,  we  have 
— x  =  — a-\-b. 

If  we  now  multiply  both  members  of  this  result  by  — 1,  we  shall 
have 

x  =  a — b  $ 

and  by  comparing  this  last  result  with  the  given  equation,  we 
observe  that  -\-a  has  been  removed  from  the  first  to  the  second 
member,  but  the  signs  of  both  the  other  terms  of  the  equation  have 
been  changed. 

Hence,  for  changing  the  sign  or  place  of  any  term  of  an  equation 
we  have  the  following 

RULE.  I.  Any  term  may  be  transposed  from  one  member  of  an 
equation  to  the  other  by  changing  its  sign;  (1,  2). 

II.  Any  term  may  be  transposed  without  changing  its  sign,  pro- 
vided the  signs  of  all  the  other  terms  be  changed ;   (3). 

III.  The  sign  of  any  term  may  be  changed  without  transposition, 
by  changing  the  signs  of  aU  the  terms  simultaneously  ;  (3). 


TRANSFORMATION.  87 


EXAMPLES     FOR     PRACTICE. 

In  the  following  equations,  transpose  the  unknown  terms  to  the 
first  member,  and  the  known  terms  to  the  second,  by  (I). 

1.  a'x-^-lc  —  ab — 2ax.  Ans.  a*x^-2ax  =  ab — be. 

2.  362— 2x—  5  =  3c— 5ax— dx. 

Ans.  5ax+dx—2x  =  3c— 3&'+5. 

3.  4c5z— a-|-35  =  x — al— 2cx. 

Ans.  4iC*x — x-}-2cx  =  a — 36 — ab. 

4.  5aZ>* — x-\-£cd  =  ax — cx-{-a*. 

Ans:  ex — ax — x  =  a3 — 5ai* — 4crf. 

In  the  following,  transpose  the  unknown  terms  to  the  first  mem- 
ber, and  the  known  to  the  second,  by  (II). 

5.  ax-\-bc  =  a*+c*x.  Ans.  c*x — ax  =  bc — a3. 

6.  4c£^ — a*x — 3cm  =  ax — m*. 

Ans.  a*x-{-ax  =  4ceZ* — 3cm-j-w8. 

7.  ax — 7-|-5ceZ  =  bc-\-a*cx — 4m9. 

Ans.  a* ex — ax  =  5cd — 7 — &c-f-4ma. 

8.  a9— c*x— Sdx  =  c*d>x— W. 

Ans.  c'cPx-}- <?x+ Zdx  =  a'-f  56a. 


CASE  n. 

151.  To  clear  an  equation  of  fractions. 

We  have  seen  (135,  2),  that  if  a  fraction  be  multiplied  by  any 
multiple  of  its  denominator,  the  product  will  be  entire ;  consequent- 
ly, if  several  fractions  be  multiplied  by  a  common  multiple  of  their 
denominators,  all  the  products  will  be  entire. 

Let  us  take  the  equation, 

^._^_19 

10       15  ~ 

Multiplying  every  term  by  30,  which  is  the  least  common  multiple 
of  the  denominators,  we  have 

9z-4z  =  360, 
in  which  all  the  terms  are  entire. 


88  SIMPLE   EQUATIONS. 

Again,  let  ^ 

a 


ab*  "   a*b 

Multiplying  every  term  by  aa&a,  observing  that  the  product 
obtained  from  the  second  fraction  is  to  be  subtracted,  we  have 

b*x  —  ax-\-ac  =.  bx-\-bc. 
Hence  the  following 

RULE.  Multiply  all  the  terms  of  the  equation  by  the  least  common 
multiple  of  the  denominators,  observing  that  when  a  fraction  has  tin 
minus  sign  before  it,  the  signs  of  the  terms  derived  from  its  numera- 
tor must  be  changed. 

NOTES.  1.  The  pupil  should  observe  that  in  multiplying  any  fraction 
it  will  be  most  convenient  to  divide  the  multiplier  by  the  denominator 
and  multiply  the  numerator  by  the  quotient. 

2.  It  will  be  obvious,  also,  that  the  equation  will  be  cleared  of  fractions; 
by  multiplying  by  the  several  denominators,  successively. 

EXAMPLES   FOR  PRACTICE. 

Clear  the  following  equations  of  fractions  : 

AM.  Gx+Kx—  9x  =  120. 


Ans.  ax-}-c?-\-cx  —  ac  =  d. 


x — a      x-\-a       x* — a* 


x — a      2x—3a 
c  aca 


Ans.  a*cx  —  a'c—  -2ax-|-3a'  =  c'z-f  ac*. 
ax  —  bx      ex  —  ax      bx  —  ex 
8c  lOa  4ac 

Ans.  5a*x  —  babx  —  4c*£c-j-4aca;  =  IQbx  —  lOccc. 


5. 


5x      So;      3—x  _  _ 

'    12~16  +  ~24~"  '    20 

AM.  lOOx—  45x+30—  lOx—  60x+  24  =  480. 

7.  4-  =•£+—  +  4-        **•*  = 

abc       ocx       acx   '   abx 


REDUCTION.  89 


REDUCTION  OP  SIMPLE  EQUATIONS. 

152.  The  Reduction  of  an  Equation  is  the  process  of  finding 
the  value  or  values  of  the  unknown  quantity,  or  the  roots  of  the 
equation. 

1*>3.  A  root  of  an  equation  is  said  to  be  verified,  if  the  two 
members  of  the  equation  prove  to  be  equal  after  the  root  has  been 
substituted  for  the  unknown  quantity. 

154:.  A  simple  equation  may  be  reduced,  by  transforming  it  in 
such  a  manner  that  the  unknown  quantity  shall  stand  alone,  and  con- 
stitute one  member  of  the  equation  }  the  other  member  will  then  ex- 
press the  value  of  the  unknown  quantity,  or  the  root  of  the  equation. 

Let  it  be  required  to  find  the  value  of  x  in  the  equation 


Clearing  of  fractions,  we  have 

20x—  8—  3z-f-2l  =  48+10x.  (2) 

By  transposition,  we  obtain 

20x—  3x—  lOx  =  48+  8  —  21.  (3) 

Uniting  similar  terms, 

7x  =  35.  (4) 

And  dividing  both  members  by  7,  we  have 

x  =  5.  (5) 

To  verify  this  value  of  x,  substitute  it  for  x  in  equation  (1)  ;  wo 
shall  have 

25—2       5—  7  25 

3  4  "6" 

Reducing  each  term  to  its  simplest  form,  we  obtain 

7f+J  =  4+4l; 
whence,  by  addition,  we  have 

8*  =  8t, 
and  the  value  of  x  is  therefore  verified. 

loo.  It  should  here  be  observed  that  an  equation  of  the  first 
degree,  containing  but  one  unknown  quantity,  can  not  have  more 
than  one  root.     For,  whatever  the  equation  may  be,  suppose  it  to  be 
8* 


90  SIMPLE   EQUATIONS. 

cleared  of  fractions,  and  the  unknown  terms  transposed  to  the  first 
member,  and  the  known  terms  to  the  second.  Then  if  we  represent 
the  algebraic  sum  of  the  coefficients  of  x  by  a,  and  the  second 
member  by  b,  the  equation  will  take  this  general  form : 

ax  =  b.  .(1) 

Now,  if  possible,  suppose  that  this  equation  has  two  roots,  r  and 
rf.  Then  since  every  root  must  satisfy  the  equation,  (14O),  we 
shall  have,  by  substituting  r  and  r'  successively  in  (1), 

ar  =  b,  (2) 

ar'=  b.  (3) 

whence,  by  Ax.  7,  we  shall  have 

ar  =  ar'-,  (4) 

or,  by  transposing  and  factoring, 

a(r— r')  =  0.  (5) 

But  equation  (5)  is  impossible,  since,  by  supposition,  r — r'  is  not 
zero,  and  a  is  not  zero.  Hence, 

An  equation  of  the  first  degree  can  not  have  more  than  one  root. 

lt>O.  From  these  principles  and  illustrations  we  derive  the  fol 
lowing 

RULE.  I.  If  necessary,  clear  the  equation  of  fractions,  and  per- 
form all  the  operations  indicated. 

II.  Transpose  the  unknown  terms  to   the  first  member  and  the 
knoicn  terms  to  the  second,  and  reduce  each  member  to  its  simplest 
form,  factoring,  when  necessary,   with  reference  to  the  unknown 
quantity. 

III.  Divide  both  members  by  the  coefficient  of  the  unknown  quan- 
tity, and  the  second  member  will  be  the  value  required,  or  the  root 
of  the  equation. 

The  three  principal  steps  in  the  reduction  of  a  simple  equation, 
containing  but  one  unknown  quantity,  may  be  briefly  stated  as 
follows : 

1st.  Clearing  of  fractions. 

2d.  Transposing  and- uniting  terms. 

3d.  Dividing  by  the  coefficient  of  x. 


REDUCTION.  91 


PRACTICAL  SUGGESTIONS. 

There  are  certain  cases  in  which  the  preceding  rule  may  be  mod- 
ified, with  advantage,  by  special  artifices. 

1  —  When  the  equation  contains  similar  terms,  or  fractions  having 
a  common  denominator,  these  should  be  united  as  far  as  possible 
before  clearing  of  fractions.  Thus, 

Given,  5C  =  100 


transposing  and  uniting  terms,        2x  -j  --  =    44  ; 

clearing  of  fractions,  14x  -f  &  —  7  =  308  ; 

15x  =  315  ; 
x=   21. 

2.  —  When  the  equation  contains  fractions  whose  numerators  or 
denominators  are  polynomial,  we  may  clear  the  equation  of  its  sim- 
pler denominators  first,  uniting  the  entire  quantities  at  each  step,  if 
possible.  Thus, 

multiplying  by  9,  6x-f  7  -f        ~     =  6x-fl2  ; 


transposing  and  uniting, 

&x—  |—  JL 

clearing  of  fractions,  2lx  —  39  =  lOa-f-5  ; 


whence,  by  division,  x  =  4. 

3  —  When  the  equation  contains  but  a  single  numerical  term,  we 
may  simply  indicate  the  multiplication  of  this  term,  in  the  clearing 
of  fractions,  until  the  final  step  in  the  reduction  is  reached.  Thus, 

Given  ^+f_+|L=88; 

multiplying  by  84,  2lx  +  12x+  Ix  -f-  4x  =  88  X  84  ; 

44x  =  88  X  84  ; 

dividing  by  44,  x  -    2X84; 

x  =  168. 


92  SIMPLE   EQUATIONS. 

EXAMPLES   FOR   PRACTICE. 

Find  the  value  of  x  in  each  of  the  following  equations. 

1.  7x  — 16  =  3x  —  4.  Ans.  x  =  3. 

2.  Zx  +  9  =  5*+  1.  Ans.  x  =  ±. 

3.  4x-\-7  =  x-}-2l  —  3-f-x.  Ans.  a  =  5J. 

4.  5a;-f  16  =  a  +  52.  Ans.  a  =  9. 

5.  5aa;  —  c  =  I  —  Sax.  Ans.  x  =  "*"c 

8a 

6.  ax4-  6  =  9a;  4-  c.  4ns.  *=  " 


—  9 
7. 


9. 


8 


10.  +      -    =-.  ^In..  »  =  20. 

7x-16       17x      3       3 


2   +    5   ~   8     2 

12  ^._i_?-4_^  —  ^_i_??~18 

1Q  17x— 12   5^— .16   10*—  3   6*  — 
13.  . 


3~      4       6       2 

.  a  =  16 


3^  —  11   5x-5  .  97-7^     Ans  x-Q 
14.   21  +  — jj--— g-  +  — 2—    ^a~9 


15      z  JL— .  A**.  «  = 

~21  4x— 11  ~3 

9x+20_4±-12       »  ^       aj  = 

16'   "36  5x-4   +4 

20,      36      5x+20      to      86 
"'   "25  +  25  ^  9z— 16       5  T  25 


ONE   UNKNOWN   QUANTITY.  93 

3x       x—  1  20x4-13  M  K 

18. -=6* Z—  A*.  x  =  5. 

19.  *_=?  +  *  =20-?+l°  ^..9. 

20.  f+-1  +  f+-2  =  16-?+3. 


21.   2z—: 


+15  =  — JT— 


»3f-— "-» 


23-  ^-—^V  =7-  ^•a5=5- 


24.  iirr_(i_^_r)  =  7*.  ^»,.  x  =  J- 

y    /  v 


•*/     r^  A.  ««x     |     «•  «^-  ^*  •t/^~^*X  j  •« 

25»  -T-+-l-=—  •f-^+3'  ^.^  =  1. 


a;  =  60. 


^-U?  =  130.  An*.  aj  =  120. 


-Aiw.  a;  =  120. 
a;  =  84. 
x  =  1260. 


a — c 

a-j-c 

a 
32.    43)x — 2a  =  3a5 — ftb*x.  Ans.  x  =  ^y 

aJ_Lac+&c 
=  0.    ulws.  x  =  — VV-T 


94  SIMPLE   EQUATIONS. 

34.  a'O-l)+am(a;— 2)  =  m*.  Ans.  x  =  ^^- 

b — fix  b 

35.  ax+cx+x  =  l-\ —  -  Ans.x=-—- 

c  a-f-c 


~  ,  ~       c— x      a 

36.    — = —  =  — •  Ans.  x  = 

6  d         b 


o7  ,  _   __  1 

'    a—  1  ^  i—  1       a+1       i-f-1  ™ 

2(aa—  2+&a 
38. 


c-f-1        c—  1    '   (c  —  I)1  c—  1 


39.      '•'-  +  *j-  -f  -  = 


a  b  c 

41.  1.25*— 6.125+.25*  =  .625*.  Ans.  x  =  7. 

42.  3.164*— 4.266  =  .24*+.08z.  Ans.  x  =  1.5. 

2.4x— .12      4.6*— 3.6      .64*— .048 

— 28 ' 4 = 7 '  X  = 

PROBLEMS 

PRODUCING  EQUATIONS  WHICH  CONTAIN  ONE  UNKNOWN  QUANTITY 


.  A  Problem,  in  Algebra,  is  a  question  requiring  the  values 
of  ODG  or  more  unknown  quantities  from  given  conditions. 

158.  The  Solution  of  a  problem  is  the  process  of  finding  the 
values  of  the  unknown  quantities. 

lt>9.  Every  problem  in  Algebra  contains  a  statement  of  the  rela- 
tions between  certain  known  and  certain  unknown  quantities.  When 
these  relations  are  such  as  to  furnish  one  or  more  equalities,  the 
process  of  solution  consists  in  expressing  these  equalities  algebraic- 
ally, and  in  reducing  the  equations  thus  obtained. 


ONE   UNKNOWN   QUANTITY.  95 

I  GO.  There  are  two  classes  of  problems  which  may  be  solved  by 
the  use  of  a  single  equation. 

1st.  Questions  referring  to  a  single  unknown  quantity. 

2d.  Questions  referring  to  two  or  more  unknown  quantities,  so 
related  that  when  one  is  known,  the  others  may  be  determined 
directly  by  the  given  conditions. 

The  following  are  examples  of  the  first  class  : 

1.  What  number  is  that  the  sum  of  whose  third  and  fourth  parts 
is  21? 
Let  x  represent  the  number  ;  then  by  the  conditions 


clearing  of  fractions,  4x-j-3x  =  21X^2, 

7x  =  21X12, 
whence  x  =  36,  Ans. 

2.  A  and  B  have  each  the  same  annual  income.     A's  yearly  ex- 
penses are  $800  and  B's  $1000,  and  A  saves  as  much  in  5  years  as 
B  saves  in  7  years  ;  how  much  is  the  annual  income  ? 
Let  x  =  the  income  ; 

then  x  —  800  =  A's  annual  savings  ; 

and  x—  1000=  B's         "          " 

Now  by  the  conditions  of  the  problem,  we  have 

5(z—  800)  =  7(z—  1000); 
whence  5x—  4000  =  7s—  7000, 


x  =  1500,     Ans. 

The  following  are  examples  of  the  second  class  : 

3.  Three  men  form  a  copartnership  with  a  joint  capital  of  $7200. 
A  put  in  a  certain  sum,  B  put  in  three  times  as  much  as  A,  and  C  put 
iu  as  much  as  both  A  and  B  ;  how  much  did  each  nian  furnish  ? 

Let  x  =  A'?  share  ; 

then  3*  =  B's      " 

and  4x  =  C's      " 

By  the  conditions,  Sx  =  $7200 

whence,  x  =  $900,  A's  share, 

Bx  =  $2700,  B's  share, 

4x  =  $3600,  C's  share. 


96  SIMPLE   EQUATIONS. 

4.  There  are  two  numbers  whose  difference  is  6 ;  and  if  J  of  the 
less  be  added  to  ^  of  the  greater,  the  sum  will  be  equal  to  $  of  the 
greater  diminished  by  £  of  the  less.     Required  the  two  numbers. 
Let  x  =  the  less ;  then  x-{-G  =  the  greater. 

By  the  conditions  of  the  problem, 

x       x-\-Q      ce-j-6      x 
3"^   ~~5~  =~~3          5> 

clearing  of  fractions,  5x+Sx+l8  =  5x+30— 3x; 
whence,  6x  =  12 ; 

x  =  2,  the  less, 
^  oj-j-6  =  8,  the  greater. 

These  examples  illustrate  the  three  essential  steps  in  the  solution 
of  any  problem  requiring  but  one  equation;  and  we  may  derive 
from  them  the  following 

GENERAL  RULE. 

I.  Represent  one  of  the  unknown  quantities  by  some  letter  or 
symbol,  and  then  from  the  given  relations  find  an  algebraic  expres- 
sion for  each  of  the  other  unknown  quantities,  if  any,  involved  in 
the  question. 

II.  Form  an  equation  from  some  condition,  expressed  or  implied, 
by  indicating  the  operations  necessary  to  verify  the  value  of  the 
unknown  quantity  represented  by  the  symbol. 

III.  Reduce  the  equation  thus  derived. 

The  three  steps  in  this  process  may  be  named  as  follows : 
1st.  The  notation. 
2d.  The  equation. 
3d.  The  reduction, 

REMARKS. 

1 — By  the  first  two  steps,  the  conditions  of  the  problem  are 
translated  from  common  into  algebraic  language.  This  is  called  the 
Ktatement  of  the  question. 

2. — The  chief  difficulty  in  the  solution  of  a  problem  is  generally 
experienced  in  obtaining  the  statement.  This  arises  in  part  from 


ONE    UNKNOWN    QUANTITY.  97 

the  fact  that  among  the  problems  which  may  be  proposed,  there  exists 
an  indefinite  variety  of  conditions ;  and  the  operator  is  left  very 
much  to  his  ingenuity,  both  in  adopting  suitable  notation  for  any 
problem,  and  in  deriving  the  equation. 

3- — Algebraic  problems  present  two  kinds  of  conditions, — Explic- 
it and  Implicit.  An  explicit  condition  is  one  which  is  distinctly 
and  formally  expressed  in  the  language  of  the  problem.  An 
implicit  condition  is  one  which  is  not  directly  expressed,  but  only 
implied,  or  left  to  be  inferred  from  other  conditions. 

4. — In  any  problem  there  are  always  as  many  conditions  as  there 
are  quantities  to  be  determined.  And  if  we  represent  one  of  the 
unknown  quantities  by  an  arbitrary  symbol,  and  then  proceed  to 
derive  expressions  for  the  other  unknown  quantities,  if  any,  each 
from  a  separate  condition,  there  will  always  remain  a  final  condition, 
cither  explicit  or  implicit,  from  which  to  derive  the  equation. 

PROBLEMS    FOE    SOLUTION. 

1.  What  number  is  that  from  which  if  6  be  subtracted,  and 
the  remainder  multiplied  by  11,  the  product  will  be  121  ? 

Ans.  17. 

2.  A  man  holds  a  lease  for  20  years,  and  J  of  the  time  past  is 
equal  to  ^  of  the  time  to  come  ;  how  much  of  the  time  has  passed  ? 

Ans.  12  years. 

8.  What  number  is  that  which  being  increased  by  4,  ^  and  j  of 
itself  is  equal  to  250  ?  Ans.  120. 

4.  Divide  77  into  two  such  parts  that  if  one  part  be  divided  by 
7  and  the  other  by  3,  the  sum  of  the  quotients  shall  be  15. 

Ans.  56  and  21. 

5.  The  sum  of  two  numbers  is  75,  and  their  difference  is  equal 
to  ^  of  the  greater ;  what  are  the  numbers  ?        Ans.  45  and  30. 

6.  After  paying  away  |  and  i  of  my  money,  I  had  $66  left ;  how 

much  had  I  at  first  ?  An*.  $120. 

^4 

7.  After  paying  away  £  of  my  money,  and  \  of  what  remaned, 

and  losing  i  of  what  was  then  left,  I  still  had  $24 ;  how  much  had 
I  at  first  ?  Ans.  $60 

9 


98  SIMPLE    EQUATIONS. 

8.  What  number  is  that  from  which  if  5  be  subtracted,  f  of  the 
remainder  will  be  40  ?  Ans.  65. 

9.  A  men  sold  a  horse  and  a  chaise  for  $200,  and  4  of  the  price  of 
the  horse  was  equal  to  ^  of  the  price  of  the  chaise.     What  was  the 
price  of  each?  Ans.  Chaise,  $120;  Horse,  CSO. 

10.  Divide  48  into  two  such  parts,  that  if  the  less  be  divided  b/ 
4,  aiid  the  greater  by  6,  the  sum  of  the  quotients  will  be  9. 

Ans.   12  and  36. 

11.  An  estate  was  divided  among  4   children  in  the  following 
manner :     The  first  received  $200  more  than  |  of  the  whole,  tho 
second  $340  more  than  1  of  the  whole,  the  third  $300  more  than  £ 
of  the  whole,  and  the  fourth  $400  more  than  |  of  the  whc-e ;  what 
was  the  value  of  the  estate  ?  Ans.  $4800. 

12.  What  number  is  that  from  which  if  91  be  subtracted,  ^  of 
the  remainder  will  be  equal  to  -Jg  of  the  number  ?         Ans.  130. 

13.  Four  men  take  stock  in  a  railroad  company,  amounting  in 
the  aggregate  to  $73,500.     A  takes  a  certain  sum,  B  takes  three 
times  as  much  as  A,  C  takes  three  times  as  much  as  A  and  B  togeth- 
er, and  D  takes  one  third  as  much  as  B  and  0.     How  much  of  the 
stock  does  A  take  ?  Ans.  $3500. 

14.  Divide  105  into  two  parts  which  shall  be  to  each  other  as 
3  to  4. 

Since  the  parts  are  to  each  other  as  3  to  4,  let 

3x  =  the  less  part ;  then  4x  =  the  greater ; 
whence  7x  =  105,  x  =  15 ; 

3x  =  45,  the  less ; 
4;e  m  60,  the  greater. 

15.  A  and  B  shared  between  themselves  a  bequest  of  $2000,  in 
the  proportion  of  7  to  9.     How  much  did  each  receive  ? 

Ans.  A,  $875;  B,  $1125. 

16.  A  farmer  made  a  mixture  of  rye,  oats,  and  peas,  using  3  bush 
els  of  rye  as  often  as  4  of  oats  and  5  of  peas      The  whole   amount 
of  grain  used  was  72  bushels;  how  many  bushels  were  there  of  each 
kind? 

Ans.  Rye,  18  bushels;  Oats,  24  bushels;  Peas,  30  bushels. 


ONE    UNKNOWN    QUANTITY.  99 

17.  From  two  casks  of  equal  size  were  drawn  quantities  which  are 
in  the  proportion  of  6  to  7  ;  and  it  appears  that  if  16  gallons  less 
had  been  drawn   from  that  which  now  contains  the  less,  only  one 
half  as  much  would  have  been  drawn  from  it  as  from  the  other. 
How  many  gallons  were  drawn  from  each  ?          ,4ns.  24  and  28. 

18.  It  is  required  to  divide  the  number  204  into  two  such  parts, 
that  §  of  the  less  being  taken  from  the  greater,  the  remainder  will 
be  equal  to  |  of  the  greater  subtracted  from  4  times  the  less. 

Ans.  154  and  50. 

19.  A  man  bought  a  horse  and  chaise  for  341  dollars.     Now  if 
|  of  the  price  of  the  horse  be  subtracted  from  twice  the  price  of  the 
chaise,  the  remainder  will  be  the  same  as  if  |  of  the  price  of  the 
chaise  be  subtracted  from  3  times  the  price  of  the  horse.     Required 
the  price  of  each.  Ans.  Horse,  $152  ;  Chaise,  $189. 

20.  A  certain  sum  of  money  was  put  at  simple  interest ;  in  8 
months  it  amounted  to  $1488,  and  in  15  months  it  amounted  to 
$1530.     What  was  the  sum  at  interest? 

It  is  often  convenient,  in  the  solution  of  a  problem,  to  avoid  the 
multiplication  of  large  numerals.  This  may  be  done  by  represent- 
ing  a  given  number  by  a  letter,  as  follows  : 

Put  a  =  1488  ;  then  a-f-42  =  1530. 
Let  x  =  the  sum  at  interest ; 

then  a — x  =  interest  for  8  months ; 

and  a+42— x  =        «      «  15         " 

Equating  two  expressions  for  the  monthly  interest, 
a — x     a-j-42 — x  f 
~8~        ~T5~ 
whence  15o— 15;c  =  8a+336— 8x, 

lx=la— 336, 
and  x  =  a— 48  =  $1440,     Ans. 

21.  A  prize  having  been  captured  by  a  privateer,  the  sum  of 
$7560  was  awarded  to  the  officers,   and  the  residue  was  divided 
equally  among  the  crew,  consisting  of  27  men.     If  the  officers  had 
received  $9560,  and  the  crew  had  consisted  of  25  men,  each  private 
would  have  received  the  same  sum  for  his  share  ;  what  was  the  value 
of  the  prize?  AUK.  $34560. ' 


100  SIMPLE    EQUATIONS. 

22.  A  .merchan t  allows  $1000  per  annum  for  the  expenses  of  his 
family,  and  annually  increases .  that  part  of  his  capital  which  is  not 
so  expended  by  a  third  of  it ;  at  the  end  of  three  years  his  original 
stock  is  doubled.     What  had  he  at  first  ?  Ans    $14,800. 

23.  A  man  having  a  lease  for  99  years,  was  asked  how  much  of 
it  had  already  expired ;  he  answered  that  f  of  the  time  past  was  equal 
to  |  of  the  time  to  come.     Required  the  time  past  and  the  time  to 
come.  Ans.  Time  past,  54  years ;  time  to  come,  45  years. 

24.  In  the  composition  of  a  quantity  of  gunpowder,  the  niter  was 
10  pounds  more  than  |  of  the  whole,  the  sulphur  was  4^  pounds 
less  than  £  of  the  whole,  and  the  charcoal  was  2  pounds  less  than 
4  of  the  niter.     What  was  the  amount  of  the  gunpowder  ? 

Ans.  69  pounds. 

25.  Divide  $183  between  two  men,  so  that  ^  of  what  the  first  re- 
ceives shall  be  equal  to  T30  of  what  the  second  receives. 

Ans.  1st,  $63  ;  2d,  $120. 

'26.  Divide  the  number  68  into  two  such  parts,  that  the  differ- 
ence between  the  greater  and  84  shall  be  equal  to  3  times  the 
difference  between  the  less  and  40.  Ans.  Greater,  42  •  Less,  26. 

27.  Four  places  are  situated  in  the  order  of  the  letters  A,  B,  C, 
D.     The  distance  from  A  to  D  is  34  miles ;  the  distance  from  A  to 
B  is  to  the  distance  from  C  to  I)  as  2  to  3 ;  and  ^  of  the  distance 
from  A  to  B,  added  to  one  half  the  distance  from  C  to  D,  is  three 
times  the  distance  from  B  to  C.     What  are  the  respective  distances? 

Ans.  12,  4,  and  18  miles. 

28.  A  man  driving  a  flock  of   sheep  to  market,  was  met  by  a 
party  of  soldiers,  who  plundered  him  of  j  of  his  flock  and  6  more. 
Afterward  he  was  met  by  another  company,  who  took  ^  of  what  he 
then  had   and  1 0  more ;  after  that  he  had  but  2  left.     How  many 
had  he  at  first  ?  Ans.  45. 

29.  A  boy  engaged  to  carry  100  glass  vessels  to  a  certain  place, 
and  to  receive  3  cents  for  every  one  he  delivered,  and  to  forfeit  9 
cents  for  every  one  he  broke.     On  settlement,  he  received  240  cents; 
how  many  vessels  did  he  break  ?  -4ns.  5. 

BO.  A  person's  entire  indebtedness  to  A,  B,  and  C,  was  $:?70.     His 


ONE    UEKXOWK 

indebtedness  to  B  was  twice  as  much  as  to  A,  and  his  indebtedness 
to  C  was  twice  as  much  as  to  A  an,d  13.  How  much  did  he  owe 
each  ?  Ant.  A,  $30  ;  B,  $60  ;  C,  Si 80. 

81.  A  company  of  4  laborers  received  $315.  B  received  li 
times  as  much  as  A,  C  received  1^  times  as  rfiuch  as  A  and  B, 
and  D  received  1|  times  as  much  as  A,  B,  and  C.  What  did  each 
laborer  receive  ?  Ans.  A,  $24  ;  B,  $36  ;  C,  $80  ;  D,  $175. 

NOTE. — Let  Gx  represent  A's  share,  and  9z  B's  share. 

32.  A  gamester,  after  losing  i  of  his  money,  won  4  shillings  ;  he 
then  lost  \  of  what  he  had,  and  afterward  won  3  shillings ;  he  then 
lost  ^  of  what  he  had,  and  found  that  he  had  only  20  shillings  re* 
maining.     How  much  had  he  at  first?  .    Ants.  30  shillings. 

33.  A  gentleman  spends  |  of  his  yearly  income  for  the  support  of 
his  family,  and  |  of  the  remainder  in  improvements  on  his  premises, 
and  lays  by  $70  a  year.     What  is  his  income  ?  Ans.  $630. 

34.  Divide  the  number  60  into  two  such  parts,  that  the  product 
of  the  two  parts  may  be  equal  to  3  times  the  square  of  the  less  part. 

Ans.  15  and  45. 

35.  My  horse  and  saddle  are  together  worth  90   dollars,  and  my 
horse  is  worth  8  times  the  value  of  my  saddle.     What  is  the  value 
of  each  ?  Ans.  Saddle,  $10 ;  Horse,  $80. 

36.  Divide  8462  between  two  persons,  so  that  for  every  dime  which 
one  receives,  the  other  may  receive  a  dollar.     Ans.  $42  and  $420. 

37.  The  rent  of  an  estate  is  8  per  cent,  greater  this  year  than  last. 
This  year  it  is  1890  dollars ;  what  was  it  last  year  ? 

AM.  $1750. 

38.  The  sum  of  two  numbers  is  840,  and  their  difference  is  equal 
to  ^  of  the  greater.     What  are  the  numbers  ? 

Ans.  504  and  336. 

39.  A  person,  after  spending   100   dollars  more  than   A  of   his 
income,  had  remaining  35  dollars  more  than  £  of  it.      Required  his 
income.  Ans.  $450. 

40.  Divide  $1520  among  A,  B,  and  C,  so  thatB  shall  have  $100 
more  than  A,  and  C  $270  more  than  B. 

Ana.  A,  $350  ;  B,  $450  ;  C,  $720. 
9* 


102  SIMPLE   EQUATIONS. 

41.  A  and  B  htive  the  same  income.     A  contracts  an  annual  debt 
amounting  to  ±  of   it ;    B  lives,  upon  J  of   it ;    at  the   end  of   two 
years  B  lends  to  A  enough  to  pay  off   his  debts,  and  has  82  dollars 
to  spare.     What  is  the  income  of  each  ?  Am.  $280. 

42.  A  sets  out  from  a  certain  place,  and  travels  at  the  rate  of  7 
miles  in  5  hours ;  and  8  hours  afterward  B  sets  out  from  the  same 
place  in  pursuit,  at  the  rate  of  5  miles  in  3  hours.     How  long  and 
how  far  must  B  travel  before  he  overtakes  A  ? 

Ans.  42  hours ;  70  miles. 

43.  A  can  perform  a  certain  piece  of  work  in  8  days,  and  B  can 
do  the  same  in  12   days ;  in  how  many   days  can   both,  working 
together,  do  it?  ARK.  4J. 

44.  A  person  has  just  6  hours  at  his  disposal ;  how  far  may  he 
ride  in  a  coach  which  travels  8   miles  an  hour,  that  he  may  return 
home  in  time,  walking  back  at  the  rate  of  4  miles  an  hour  ? 

Ans.  16  miles. 

45.  A  can  dig  a  trench  in  one  half  the  time  required  by  B,  B 
can  dig  it  in  two  thirds  of  the  time  required  by  C,  and  all  together 
can  dig  it  in  6  days ;  find  the  time  that  each  alone  would  require. 

Ans.  A,  11  days;  B,  22  days;  C,  33  days. 

46.  A  and  B  start  from  opposite  points  and  travel  toward  each 
other,  A  at  the  rate  of  3  miles  an  hour,  and  B  at  the  rate  of  4 
miles  an  hour.     At  the  same  time,  C  sets  out  with  A  and  travels  at 
the  rate  of  5  miles  an  hour.     After  meeting  B  he  turns  back  and 
travels  until  he  meets  A  ;  he  then  finds  that  the  whole  time  elapsed 
since  starting  is  10  hours.     How  far  apart  were  A  and  B  at  the 
beginning  ?  Ans.  72  miles. 

47.  Two  farmers  owning  a  flock  of  sheep,  agree  to  divide  them. 
A  takes  72  sheep ;  B  takes  92  sheep,  and  pays  A  $35.     Required 
the  value  of  the  flock.  Ans.  $574. 

48.  A  crew  which  can  row  at  the  rate  of  12  miles  an  hour  in 
still  water,  finds  that  it  takes  7  hours  to  come  up  a  river  a  certain 
distance,  and  5  hours  to  go  down  again.     At  what  rate  does  the 
river  flow  ?  Ans.  2  miles  per  hour. 


TWO   UNKNOWN    QUA 

SIMPLE    EQUATIONS 
CONTAINING   TWO   UNKNOWN   QUANTITIES. 

161.  TVe  have  seen  that  every  equation  containing  one  unknown 
quantity  can  be  satisfied  with  one  value,  and  only  one  value,  of  the 
unknown  quantity,  (155).  But  if  we  consider  a  single  equation 
containing  two  unknown  quantities,  we  shall  find  that  for  every 
value  which  we  please  to  give  to  one  of  the  unknown  quantities. 
we  can  determine  a  corresponding  value  of  the  other  unknown 
quantity,  such  that  the  set  of  values  will  satisfy  the  equation. 

Thus,  let 

=  17.  (1) 


Put  x  =  1,  and  substitute  this  value  in  the  given  equation  ;  we 
have 

24-3y  =  17, 

>w& 

Now  the  set  of  values,  x  —  1,  y  —  5,  will  satisfy  the  equation ; 
for,  by  substitution,  we  have 

2+15  =  17. 

In  the  same  manner,  we  may  obtain  the  following  sets  of  values, 
each  one  of  which  will  satisfy  equation  (1) : 

1.  x=l,    y  =  5. 

2.  a  =  2,    y  =  4f 

3.  *  =  3,    y  =  3§. 

4.  a  =  4,    y  =  3. 

It  is  evident  that  there  is  no  limit  to  the  number  of  sets  of  val- 
ues that  may  be  obtained.  The  equation,  and  also  the  quantities, 
in  such  cases,  are  said  to  be  indeterminate.  Hence, 

169.  An  Indeterminate  Equation  is  one  which  is  satisfied  by 
an  infinite  number  of  values  of  the  unknown  quantities. 

Every  sinyle  equation  containing  two  unknown  quantities,  is 
indeterminate. 

163.  If  we  take  two  equations  with  two  unknown  quantities,  as 

=  31,  (1) 

2ij  =  19,  (2) 

it  is  evident  that  we  may  obtain  as  many  sets  of  values  as  we  please, 


104  SIMPLE    EQUATIONS. 

winch  will  satisfy  each  equation,  considered  separately.     Thus,  pro- 
ceeding as  before,  we  find  that  the  set, 

x  =  5,     y  =  41, 
will  satisfy  the  first  equation  ;  and  a  different  set, 

x  =  4,    y  =  3i, 
will  satisfy  the  second  equation. 

Now  suppose  we  are  required  to  satisfy  Loth  equations  with  the 
same  set  of  values  for  x  and  y. 

Multiplying  (1)  by  3,  and  (2)  by  2,  we  have 

6-r-f  15.y  =  93,  (3) 

6x+  4y  =  38.  (4) 

Subtracting  (4)  from  (3),  we  have 

lly  =  55,  (5) 

whence  y  —  5.  (6) 

Substituting  this  value  of  y  in  (1),  we  have 

2z+25  =  31,  (7) 

whence,  x  =  3.  (8) 

Thus  we  have  a  single  set  of  values, 

x  =  3,  y  =  5, 

which  will  satisfy  both  equations.     For,  let  these  values  be  substi- 
tuted in  the  given  equations  j  we  shall  have 
6-4-25  =  31, 
9+10  =  19. 
Equations  thus  related  are  said  to  be  simultaneous.     Hence, 

104.  Simultaneous  Equations  are  those  which  must  be  satisfied 
by  the  same  values  of  the  unknown  quantities  which  enter  them. 

When  two  or  more  simultaneous  equations  are  given,  the  values 
of  the  unknown  quantities  are  determined  by  a  process  called 

ELIMINATION. 

1O«5.  Elimination  is  the  process  of  combining  equations  in  such 
a  manner  as  to  cause  one  or  more  of  the  unknown  quantities  con- 
tained in  them  to  disappear. 

There  are  four  principal  methods  of  elimination ; 

1st,  By  addition  and  subtraction ;  2d,  By  comparison ;  3d,  By 
substitution ;  4th,  By  indeterminate  multipliers. 


TWO   UNKNOWN   QUANTITIES.  105 


CASE  I. 

166.  Elimination  by  addition  and  subtraction. 

I.  Given  3z-f  2y  =  23,  and  4x— 3y  =  8,  to  find  the  values  of  x 
d^r. 

OPERATION. 

23;  (1) 

8.  (2) 

Multiplying  equation  (1)  by  4,  and  equation  (2)  by  3,  we  have 

I2x+8y  =  92,  (3) 

12*— 9y  =  24;  (4) 

subtracting  (4)  from  (3),  17y  =  68;  (5) 

whence,  y  =  4.  (6)  , 

Thus,  we  have  eliminated  x,  and  found  the  value  of  ^. 
Again,  multiplying  equation  (1)  by  3,  and  equation  (2)  by  2,  we  have 

9*+6,y  =  69,  (7) 

8**— 6y  =  16  ;  (8) 

adding  (7)  to  (8),  I7x  =  85  ; 

whence,  x  =    5. 

We  have  thus  eliminated  y,  and  found  the  value  of  x.     Hence, 
RULE.  I    Multiply  or  divide  the  equations   by  such  numbers  or 
quantities  that  the  coefficients  of  the  quantity  to  be  eliminated  shall 
be  made  equal  in  the  two  equations. 

II.  If  these  coefficient*  have  like  signs,  subtract  one  of  the  prepared 
equations  from  the  other,  member  from  member;  if  they  have  unlike 
signs,  add  the  equations,  member  to  member. 

NOTES.  1.  In  preparing  the  given  equations  by  multiplication,  find  the 
least  common  multiple  of  the  coefficients  of  the  letter  to  be  eliminated, 
and  divide  this  multiple  by  each  coefficient ;  the  quotients  will  be  the  least 
multiplies  that  can  be  used.  If  the  coefficients  are  prime  to  each  other, 
it  is  evident  that  each  equation  must  be  multiplied  by  the  coefficient  in 
the  other  equation. 

2.  It  is  generally  convenient  to  clear  the  equations  of  fractions,  before 
applying  the  rule.  This  is  not  necessary,  however.  For  if  any  letter  has 
fractional  coefficients  in  the  two  equations,  the  fractions  may  be  reduced 
to  a  common  denominator  ;  it  will  then  be  necessary  to  render  the  nume- 
rators equal  by  multiplication  or  division,  according  to  the  rule. 


106  SIMPLE    EQUATIONS, 


CASE  II. 


167.  Elimination  by  comparison. 

1.  Given  ox-\-5y  =  42,  and  2x+y  =  14,  to  find  the  values  of 
x  and  y. 

OPERATION. 


From  (1),  by  transposition,  etc., 
«     (2), 

therefore,  by  Ax.  7, 

clear! ng  of  fractions,  84 — lOy  =  42 — 3y ; 

whence,  ly  =  42, 

y—    6. 
Substituting  the  value  of  y  in  (3),  x  =    4. 

Hence,  we  have  the  following 

RULE.  I.  Find  the  value  of  the  same  unknown  quantity,  in  terms 
of  the  other,  from  each  of  the  given  equations. 

II.  Form  an  equation  by  placing  these  two  values  equal  to  each 

other. 

CASE  in. 

,  168.  Elimination  by  substitution. 

1.  Given  Zx+2y  =  16,  and  bx—Zy  =  14,  to  find  the  values 
3f  :c  and  y. 

OPERATION. 

Zx+2y  =  16.  (1) 

5x— 3y  =  14.  (2) 

From  (1),  we  obtain  y  =.  —          .  /g\ 

2       ' 


TWO    UNKNOWN    QUANTITIES.  107 

Substituting  this  value  of  y  in  equation  (2); 

,        48-9* 

5* __  =  14; 

clearing  of  fractions,  10x — 48-f9x  =±  28, 

whence,  x  =  4. 

'  < 

Substituting  the  value  of  cc  in  (3)}  ^  =  2. 

Hence  the  following 

RULE.  I.  Find  the  value  of  one  of  the  unknown  quantities,  in 
terms  of  the  other,  from  either  of  the  given  equations. 

II.  Substitute  this  value  for  the  same  unknown  quantity  in  the 
other  equation. 


CASE   IV. 

169.  Elimination  by  indeterminate  multipliers. 

1.  Given  2x-f  3y  —  23,  and  5cc-j-2#  =  30,  to  find  the  values  of 
x  and  y. 

If  we  multiply  the  first  equation  by  a  quantity,  m,  which  is  as 
yet  undetermined,  we  have 

2mx+3my  =  23m,  (1) 

5*+2y  =  30.  (2) 

Subtracting  (2)  from  (1),  and  factoring  with  reference  to  x  and  y, 
we  have 

(2m— 5)x-f  (3m— 2)y  =  23m— 30.  (3) 

It  is  evident  that  equation  (3)  is  true,  whatever  be  the  value  of 
m.  We  may  therefore  assume  m  to  be  of  such  a  value  that  the  co- 
efficient of  one  of  the  unknown  quantities  shall  become  zero ;  this 
will  eliminate  that  unknown  quantity,  by  causing  the  term  contain- 
ing it  to  disappear.  Thus,  assume 

2m— 5  =  0,  (4) 

whence,  m  =  f .  (5) 

But  if  2m — 5  =-0,  the  first  term  of  (3)  is  0,  and  that  equation 
becomes 

(3m — 2)#  =  23m — 30  ;  (6) 

23m— 30 


108  SIMPLE    EQUATIONS. 

If  we  now  substitute  in  (7)  the  value  of  m,  as  given  in  (5)r  wo 
shall  have 

.  23x^—30  _  115—60  __  55 
:    3x|— 2    :       15-4       -TI  = 

In  like  manner  we  may  eliminate  x  from  (3).  To  accomplish  this, 
assume 

3m— 2  =  0,  (9) 

whence,  m  =  |.  (10) 

But  if  3m — 2  =  0,  equation  (3)  becomes 

(2m— 5)a;  =  23m— 30  ;  (11) 

23m— 30 

whence,  x  =  — —  (12) 

2m — o 

Substituting  in  (12)  the  value  of  m,  as  expressed  in  (10),  we  have 
23XI-30  _  46-90  _ -44 

~2xf^  •  1=15    =n- 

This  method  of  elimination  is  due  to  the  French  writer,  Bezoufr. 
It  is  called  the  method  of  indeterminate  multipliers,  because  the 
multipliers  which  we  employ  are  at  first  undetermined.  Strictly 
speaking,  the  multipliers  thus  used  are  not  indeterminate  ;  for,  in 
order  to  effect  the  elimination  of  the  unknown  quantities,  they  must 
have  certain  definite  values,  which  values  are  always  determined  in 
the  course  of  the  operation. 

Recurring  to  the  operation  above,  we  notice  that  m  has  two  values, 
thus : 

»*  =  *,  (1) 

m  =  |.  (2) 

The  first  value  of  m  is  the  one  by  which  x  was  eliminated ;  tho 
second,  the  one  by  which  y  was  eliminated.  Resume  the  given 
equations, 

2x+3y  =  23,  (1) 

5ar+2y  =  30.  (2) 

It  will  be  readily  seen  that  if  the  first  equation  be  multiplied  by 
the  first  value  of  m,  the  coefficients  of  x  will  be  alike  in  the  two 
equations ;  and  if  the  first  equation  be  multiplied  by  the  second 
value  of  m,  the  coefficients  of  y  will  be  alike  in  the  two  equations. 
It  is  obvious,  therefore,  that  the  method  of  indeterminate  multipliers 
is  but  a  modification  of  the  method  «f  addition  and  subtraction. 


TWO    UNKNOWN    QUANTITIES.  109 

Recurring  ngnin  to  the  above  operation,  it  is  evident  that  if  the 
sum  instead  of  the  difference,  of  equations  (1)  and  (2)  had  been  taken, 
the  elimination  might  have  been  effected  with  equal  facility.  Hence, 

RULE.  I.  Multiply  one  of  the  given  equations  by  the  indefinite 
factor  m,  and  then  take  the  sum  or  difference  of  this  result  and  the 
other  equation,  fcrc for iny  with  reference  to  the  unknown  quantities. 

II.  Put  the  coefficient  of  one  of  the  unknown  quantities  in  this 
last  equation  equal  to  zero,  and  determine  the  value  of  m;  then 
substitute  this  value  in  the  equation  Containing  m,  and  the  result  will 
be  an  equation  of  but  one  unknown  quantity. 

17O.  In  the  reduction  of  simultaneous  equations  containing  two 
unknown  quantities,  sometimes  one  of  the  preceding  methods  of 
elimination  is  preferable,  and  sometimes  another,  according  to  the 
special  relations  of  the  coefficients. 

EXAMPLES   FOR   PRACTICE. 


Ans.  x  =  6  ;  y  =  4. 

2.  Given    \    6x+2y  =  19    I    to  find*  and  y. 

|    7x_6y=   9    f 

Ans.  x  =  3  ;  y  =  2. 

3.  Given     j    3*+^  =  ™    I    to'  find  x  and  y. 

(      *4-4y  =  38    j 

Ans.  x  =  10  ;  y  =  7. 

4.  Given    \    5x-3y  =  36    )        ^         d 

I    2ic+9y  =  96    f 

Ans.  x  =  12  ;  y  =  8. 

6-  GiTen  {  sES^S  }  *»«-*•»*»• 

^Lws.  a;  =  20  ;  y  =  2. 

6.    Given    {    &*-4y=     40    )        fid         d 
I      a;—  5,y  =  —  97    J 

=  28  ;  y  =  25. 


iven    j    ^+J^=     °    1    to  find  »  and  y. 
(    Qx—  I2    =  —  I    J 


7.    Given 

Qx—  I2y 


110  SIMPLE    EQUATIONS. 


8.  Given    j    7x+7j/  =  30    )    to  ^  x  and  y> 

(    3x-f-4y  =  17    ) 

Ans.  x  =  4  ;  y  =  4J. 

9.  Given    I    ?Z+^  =  «    1    to  find*  and  y. 

(    5x  —  6y  =  55    j 

Ana.  x  =  5  ;  y  =  —  5. 

10.  Given    J    ^"^       *    [    to  find  x  and  y. 

(    lOx-fty  =  —  43    ) 

^Liu.  x  =  —  2i;y  =  —  5J. 

11.  Given    j    ^~^=      ^    [    to  find,  and  y. 

(   2x  —  3y  =  —  450    ) 


Ans.  x  =  300 ;  y  =  350. 

f    x       v  *\ 

V —  =  20    i 

A    2       4  r 

12.   Given 


to  find  x  andy 
Ans.  x=l6:  y  =  24. 


TWO    UNKNOWN    QUANTITIES. 


Ill 


17.   Given 


18.   Given 


19.   Given 


21.    Given 


.4ns.  cc  =  3  ;  y  =  7. 


4a>fl7  68 

20.    Given    <^  "^   >    to  find  a;  and  y. 

5y-f27  54 


J.TIS.  a;  =  13  ;  y  =  3. 


23.    Given 
find  x  and  y. 


73—3 
' 


to  ^n(^  x 


Ans.  x  =  21  ;  y  =  20. 


151-16. 


—  7 


112 
24. 


SIMPLE    EQUATIONS. 


.    Given    I  af  +  ^  =  djt  \  to  find  x  and  y. 
(  a'a-f&'y  ==  d'  ) 


25.    Given 


I'd—  Id' 


a'd—ad' 


to  find  x  and  y. 


cz-f  ay  = 


ac 


az-f-cy  =  -^r 
26.    Given    4  i    «  >  to  find  x  and  ^ 


a  c 

Ans.  x  =  --;y  =  - 


SBIPLE  EQUATIONS 

CONTAINING    MORE    THAN   TWO    UNKNOWN   QUANTITIES. 


171.  If  we  have  three  or  more  simultaneous  equations,  they  may 
reduced  by  successive  eliminations,  as  follows  : 

C  2x+  4y+  4^  =  18;  (i) 

n    < 

I 

Multiplying  (1)  by  3, 

(2)  by  2, 

subtracting  (5)  from  4, 

multiplying  (1)  by  5,  ' 

(3)  by  2, 

subtracting  (8)  from  (7), 
multiplying  (6)  by  4, 

(9)  by  3, 

subtracting  (11)  from  (10), 
yhence, 
substituting  value  of  a  in  (9), 

".         values  of  y  and  z  in  (1). 


3^_j_  3y_|_  2z  =  17  ; 
5*+  %+  5^  =  32. 

(2) 
(3) 

(4) 
(5) 

(6) 
(T 

(9) 
(10) 
(11) 
(12) 

6z+12y-fl2^  =  54, 
6z+  6y-f  4^  =  34; 

6y+  8z  =  20; 

10a;+20y-f  20z  =  90, 
10z4-12y+102  =  64  ; 

8y+10^  =  26  ; 

24y-j-32^  =  80, 
24y-f  30^  =  78  5 

2^  =  2; 
-i  . 

9),                 y-2- 

TWO   UNKNOWN    QUANTITIES.  113 


INFERENCES. 

1. — If  we  have  n  equations,  and  proceed  as  above  to  combine  one 
of  them  with  each  of  the  others,  eliminating  the  same  letter  by 
each  combination,  we  shall  have  n — 1  derived^equations  containing 
all  of  the  unknown  quantities  except  the  one  eliminated. 

2. — If  then  we  combine  one  of  these  derived  equations  with  each 
of  the  others,  eliminating  another  letter,  we  shall  have  n — 2  derived 
equations,  containing  all  of  the  unknown  quantities  except  the  two 
eliminated  quantities. 

3. — Since  each  succeeding  group  of  derived  equations  consists  of 
one  less  equation  than  the  preceding  group,  it  follows  that  if  this 
process  of  successive  elimination  be  continued,  the  (n — l)th  group 
will  consist  of  a  single  equation ;  and  this  will  contain  all  of  the 
unknown  quantities  except  the  n — 1  eliminated  quantities.  Hence, 

4. — If  the  number  of  original  equations  equals  the  number  of  un- 
known quantities,  the  final  equation  will  contain  but  a  single  un- 
known quantity,  the  value  of  which  may  be  found.  By  substituting 
this  value  in  one  of  the  equations  of  the  preceding  group,  the  value 
of  a  second  unknown  quantity  may  be  determined;  and  so  on. 

5. — But  if  the  number  of  original  equations  is  less  than  the 
number  of  unknown  quantities,  the  final  equation  will  contain  more 
than  ©ne  unknown  quantity,  and  will  be  indeterminate,  (162);  and 
consequently,  the  given  equations  will  be  indeterminate. 

6. — In  the  solution  of  two  or  more  simultaneous  equations  of  the 
first  degree,  by  successive  eliminations,  the  value  of  each  letter 
is  determined  finally  by  a  simple  equation  containing  only  that  letter. 
And  since  every  such  equation  can  have  only  one  root,  (155),  it 
follows  that  any  group  of  simultaneous  equations  can  be  satisfied  by 
only  one  set  of  values  of  the  unknown  quantities. 

172.  From  the  foregoing  inferences  we  derive  the  following 

RULE.  I.  Combine  one  of  the  given  equations  with  each  of  tlie 
others,  eliminating  the  same  unknown  quantity  by  each  combination; 
then  combine  one  of  the  new  equations  with  each  of  the  others,  elim- 
inating a  second  unknown  quantity,  and  thus  continue  till  a  final 
equation  is  obtained,  containing  out  one  unknown  quantity. 
10*  H 


114  SIMPLE   EQUATIONS. 

II.  Reduce  this  final  equation,  and  find  the  value  of  the  unknown 
quantity  which  it  involves  ;  substitute  this  value  in  an  equation  con- 
taining two  unknown  quantities,  and  thus  find  the  value  of  a  second; 
substitute  these  values  in  an  equation  containing  three  unknown  quan- 
tities, and  find  the  value  of  a  third}  and  so  on,  till  the  values  of  all 
are  found. 

PRACTICAL  SUGGESTIONS. 

This  rule  may  be  modified  in  certain  cases,  as  follows  : 
1.  —  Instead  of  combining  the  first  equation  with  each  of  the 
others,  we  may  pursue  any  order  of  combination,  or  adopt  any  one 
of  the  four  methods  of  elimination,  which  seems  best  suited  to  tho 
mutual  relations  of  the  coefficients.  The  following  example  will  il- 
lustrate the  precept  just  given  : 

(  x+  y+  z  =    9  ;  (1) 

Given        *+ty-f-3z  =  16  ;  (2) 


Subtracting  (1)  from  (2),  y+2z  =  1  ;  (4) 

(2)  from  (3),  ?/+   z  =  5  ;  (5)   ' 

«  (5)  from  (4),  z  =  2  ; 

substituting  value  of  z  in  (5),  y  =  8  ; 

"          values  of  y  and  z  in  (1),        .JL  =  4. 

2.  —  If  two  or  more  of  the  equations,  taken  together,  contain  less 
than  all  the  unknown  quantities,  it  is  generally  most  convenient  to 
employ  these  equations  first,  in  the  process  of  elimination.     Thus, 
s  2u—x+  fy+Zz  =  19  ;  (1) 

\  3w-f5x—  4#          =  23  ;  (2) 

Given   ^  4u+  3*  =  32  ;  (3) 

^  2u+5a;  __  =30.  (4) 

Multiplying  (4)  by  2,  4u+10x  =  60  ; 

bringing  down  (3),  4w+  3x  =  32  ; 

by  subtraction,  7-r  =  28, 

x=    4; 

substituting  in  (3),  u  =    5, 

«  (2),  y  =    3, 

«          «  (1),  z  =    2. 


TWO    UNKNOWN    QUANTITIES. 


115 


3.  —  When  the  coefficients  sustain  to  each  other  relations  of  equal- 
ity or  symmetry,  it  is  often  convenient  to  employ  an  auxiliary  quan- 
tity, as  in  the  two  examples  which  follow  : 


\  u+v+x+z  =  15  ; 
1.    Given    <    u+v+y+z  =  16 ; 

(  v+x+y+z  =  18.' 
Since  in  each  equation  one  letter  is  wanting,  let 


(2) 
(3) 
(4) 
(5) 


then 


s-z  =  14, 

*— y  =  15, 
«— x  =  16, 

s— v  =  17, 
=  18. 


L-  }y  =  5, 

B     uf    <.       #  =  4, 

tion,    )  v  =  3, 

(M=2. 


Hence,  by  substitu- 
ting the   value  of  <^;  x  =  4, 
s  in  each  equatk 


By  addition,  5s  —  s  =  80, 


f 
2.    Given  < 


Assume 


equation  (1)  becomes 
«      (2)         « 
"      (3)         « 

Multiplying  (4)  by  8, 
(5)  by  4, 
"  (6)  by  3, 

Adding  (7),  (8)  and  (9), 


=  450; 
=  490. 


x+  y+  z  =  s-, 

4s—  Bx  =  300, 
7s —  6y  =  450, 
9s—  8z  =  490. 

32s— 24*  =  2400, 
28s— 24y  -  1800, 
27s— 24s  =  1470. 

87s— 24s  =  5670, 

63s  =  5670, 

s=   90. 


Substituting  value  of  s  in  (4),  x  =  20, 

(5),  #  =  30, 

"  "         "         (G),  2  =  40. 


(1) 
(2) 
(3) 


(4) 
(5) 
(C) 

(7) 
(8) 
(9) 


116  SIMPLE    EQUATIONS. 


EXAMPLES  FOB  PRACTICE. 

Required  the  values  of  the  unknown  quantities  in  the  following 
equations  : 


Ans.     -(  y  = 
6x-|-7y— 


f  2x+4y—3z  =  22  ^ 

I  6x4-  ly—  z  =  63  J 

r    3x497,48*  =  41  1 

5.    <     5x44y— 2z  =  20   V 

I  lLc47#— 6*  =  37  J 

/-  x4^4^  =  31  ^ 

j.    J    x+y-,  =  25   I 
I  x—y—z  =    9  J 

f  x4^4^  =  26  ^| 

t.    -|   x—y       =    4  V 

I  x— z        =    6  j 

/     x—y—z=    6.  j 

>.     J  3.T/— x~^  =  12    V 

(  7^_?/_a;  =  24   ) 


11x47^— Qz 

x4y4^  =  31^  (x=20, 

fy=   8, 
x — y- 

x4^4^  =  26 

}>  Ans.    \ 

z  =    6. 


x — y — z  =    6   )  /  ic  =  39, 

u4.ns.    -(^  ==  21? 
-2/ — x  =  z*   /  (  2  =  12. 

NOTE. — In  the  last  example,  assume  x+y+z  =  5,  and  add  this  equation 
to  each  of  the  given  equations.    Then  determine  s  as  hi  (3,  2). 


6. 


7. 


8          4x43y43«  =    8  Ans. 


TWO    UNKNOWN    QUANTITIES.  117 


/  n+v+x+y+2*  =  52  \ 

I      U  +  V  +  X+Z  +  2y    =T    50     I 

/  u+v+y+z+2x  =  48  \ 
}  u+x+y+z+2v  =  46  I 
(  t4.*+y+*+2«  =  44  J 


9.    /  u+v+y+z+2x  =  48   >  Ant. 


y—  2^  = 
_  04-8*  =  35 
10.  3«+  <=13  Ans. 


c—y+3t—  u  = 

v+  y—  z  =  l  - 
11.    <    8*+3y-6*  =  l    V 


r     x+  y—  z  =  l  \ 

.     <    8.r+3^—  6^  =  1    V 

(  3z—  4x—      =  l  J 


Ans. 


{ 


x=^a, 

') 


y  =  /T^ 


z  =  T'-  a. 

X  = 


7 

TT 


118  SIMPLE    EQUATIONS. 

(  lx—  By  =  a  I  I  x  =  4a, 

17.     I  5# — llx  =  a  >  Ans.    \  y  — 


a  ~*~6  ~~6       a 


ax 


_c  ,        = 

«  —  -  .  «  x»  7* 


&c  ac  ac  ai 

ex  +  y  +  02  =  2a 


20. 

acx — y-f~ac*  =  a*-f-c* 


a 

a  a?-|-a  y-f- 


,,.i 

.„.. 

i1*  =  a3  > 


PROBLEMS 

PRODUCING   EQUATIONS    CONTAINING   TWO   OR    MORE 
UNKNOWN    QUANTITIES. 

173.  Two  or  more  equations  are  said  to  be  independent,  when 
they  are  not  derived  one  from  the  other,  and  can  not  be  reduced  to 
the  same  form ;  as 

3*+  y  =  17, 
2x+8y  =  28. 

Equations  derived  from  the  same  problem  are  independent,  when 
they  express  different  conditions  of  that  problem. 


TWO    OR    MORE    UNKNOWN    QUANTITIES.  119 

We  have  seen  tbat  a  group  of  equations  will  be  determin- 
ate, when  the  number  of  equations  is  equal  to  the  number  of  un- 
known quantities,  but  not  otherwise,  (172,  4  and  5).  Hence, 

A  problem  icill  be  capable  of  solution  only  iclien  its  conditions  fur- 
nixli  as  many  independent  equations  as  there  age  unknown  symbols 
employed  in  the  notation. 

1.  Find  two  numbers,  such  that  twice  the  first  plus  three  times 
ihe  second  is  equal  to  105  ;  and  three  times  the  first  plus  twice  the 
second  is  95.  Ans.  First,  15;  Second,  25. 

2.  Find  three  numbers,  such  that  the  first  with  ^  of  the  sum  of  the 
second  and  third  shall  be  120;  the  second  with  J  of  the  sum  of  the 
third  and  first  shall  be  90;  and  the  sum  of  the  three  shall  be  190. 

Ans.  50,  65,  75. 

3.  A  sum  of  money  was  divided  among  three  persons,  A,  B,  and 
C,  as  follows :  the  share  of  A  exceeded  |  of  the  shares  of  B  and  C 
by  $120 ;  the  share  of  B  exceeded  f  of  the  shares  of  A  and  C  by 
$120;  and  the  share  of  C  exceeded  |  of  the  shares  of  A  and  B  by 
$120.     What  was  each  person's  share? 

Ans.  A's,  $600;  B's,  $480;  C's,  $360. 

4.  A  and  B,  working  together,  can  earn  $40  in  6  days ;  A  and  C 
can  earn   $54  in  9  days;  and  B  and   C  can  earn  $80  in  15  days. 
How  much  can  each  person  alone  earn  in  one  day  ? 

Ans.  A,-$3f;  B,  S3;  C,  $2i. 

5.  A  man  has  4  sons.     The  sum  of  the  ages  of  the  first,  second 
and  third  is  18  years;  the  sum  of  the  ages  of  the  first,  second  and 
fourth  is  1 6  years ;  the  sum  of  the  ages  of  the  first,  third  and  fourth 
is  14  years;  the  sum  of  the  ages  of  the  second,  third  and  fourth  is 
12  years.     What  are  their  respective  ages? 

Ans.  8,  6,  4,  and  2  years. 

6.  Three  persons  engaged  in  throwing  dice,  on  certain  conditions. 
In  the  first  game  A  forfeited  to  B  and  C,  respectively,  as  many  shil- 
lings as  each  of  them  had  ;  in  the  second  game  B  forfeited  to  A  and 
C,  respectively:  as  many  shillings  as  each  of  them  then  had;  in  the 
third  game  C  forfeited  to  A  and  B,  respectively,  as  many  shillings  as 
each  of  them  then  had;  they  had  then  16  shillings  apiece.     How 
many  shillings  had  each  at  first?  Ans.  A,  26;  B;  14;  C,  8. 


120  SIMPLE    EQUATIONS. 

7.  A  gentleman  left  a  sum  of  money  to  be  divided  among  four 
si  rvants.  so  that  the  share  of  the  first  should  be  |  of  the  sum  of  the 
shares  of  the  other  three,  the  share  of  the  second  J  of  the  sum  of 
the  other  three,  and  the  share  of  the  third  -}  of  the  sum  of  the  other 
three.     On  making  the  division,  the  fourth  had  14  dollars  less  than 
the  first.     Required  the  sum  divided,  and  the  several  shares. 

Ans.  Sum  divided,  $120;  shares,  $40,  $30,  $24  and  $26. 

8.  A  person  has  two  horses  and  two  saddles,  the   saddles  being 
worth  $15  and  $10,  respectively.      Now  the  value  of   the  better 
liorse  with  the  better  saddle  is  |  of  the  value  of  the  other  horse  and 
saddle ;  but  the  value  of  the  better  horse  with  the  poorer  saddle  is 
||  of  the  value  of  the  other  horse  and  saddle.     What  are  the  values 
of  the  two  horses  ?  Ans.  $65  and  $50. 

9.  A  vintner,  in  mixing  sherry  and  brandy,  finds  that  if  he  takes 
2  parts  of   sherry  to  1  of  brandy,  the  mixture  will  be  worth  78 
shillings  per  dozen  ;  but  if  he  takes  7  parts  of  sherry  to  2  of  brandy, 
the  mixture  will  be  worth  79  shillings  per  dozen.      What  are  the 
sherry  and  brandy  worth  per  dozen  ? 

Ans.  Sherry,  81  shillings ;  Brandy,  72  shillings. 

10.  Two  persons,  A  and  B,  can  perform  a  piece  of   work  in  16 
days.     They  work  together  for  four  days,  when  A  is  called  off,  and 
B  is  left  to  finish  it,  which  he  does  in  36  days.      In  what  time 
would  each  do  it  separately  ? 

Ans.  A,  in  24  days ;  B,  in  48  days. 

11.  What  fraction  is  that,  whose  numerator  being  doubled,  and 
denominator  increased  by  7,  the  value  becomes  f ;    but  the  denomi- 
nator being  doubled,  and  the  numerator  increased  by  2,  the  value 
becomes  f  ?  Ans.  4 . 

12.  Two  men  were  wishing  to  purchase  a  house  together,  valued 
at  240  dollars.      Says  A  to  B,  "  If  you  will  lend  me  ~  of  your  money 
I  can  purchase  the  house  alone  f  but  says  B  to  A,  "  If  you  will  lend 
me  |  of  yours,  I  can  purchase  the  house  alone."     How  much  money 
had  each?  Am.  A,  $160;  B,  $120. 

13.  A  pleasure  party,  having  chartered  a  boat  for  a  certain  sum 
found,  on  settling,   that  if '  their  number  had    been  4   more,  they 
would  have  had  a  shilling  apiece   less   to  pay  ;    but  if  their  number 


TWO    OB    MORE    UNKNOV'X    QUANTITIES.  121 

had  been  3  less,  they  would  have  had  a  shilling  apiece  more  to  pay. 
What  was  their  number,  and  what  had  each  to  pay  ? 

Ans.  24  persons;  each  paid  7  shillings. 

14.  A  certain  number  consists  of  two  places  of  figures,  units  and 
tens;  the  number  is  equal  to  4  times  the  sum  of  4ts  digits,  and  if  27 
be  added  to  the  number,  the  order  of   the  digits  will  be  inverted. 
What  is  the  number  ? 

NOTE  1.  Let  x  represent  the  digit  in  the  place  of  tens,  and  y  the  digit 
iu  place  of  units ;  then  \Qx+y  will  express  the  number. 

Ans.  8G. 

15.  A  number  is  expressed  by  three  figures  whoso  sum  is  11 ; 
the  figure  in  the  place  of  units  is  double  that  in  the  place  of  hun- 
dreds ;   and  if   297   be  added   to  the  number,  the  result  will  bo 
expressed  by  the  same  figures  with  their  order  reversed.     What  is 
the  number  ?  Ans.  326. 

1(3.  Divide  the  number  90  into  three  parts,  such  that  twice  the 
first  part  increased  by  40,  three  times  the  second  part  increased  by 
20,  and  four  times  the  third  part  increased  by  10,  may  all  be  equal 
to  one  another.  Am.  First  part,  35  ;  Second,  30  ;  Third,  25. 

17.  A  person  placed  $100,000  out  at  interest,  a  part  of  it  at  5 
per  cent.,  and  the  rest  at  4  per  cent.;  the  yearly  interest  received  on 
the  whole  was  S4G40.     Required  the  two  parts  of  the  principal. 

Ans.  $64,000  and  $36,000. 

18.  A  person  put  out  a  certain  sum  of  money  at  interest  at  a 
certain  rate.     Another  person  put  out  $10,000  more  than  the  first, 
at  a  rate  per  cent,  greater  by  1,  and  received  an  income  greater  by 
$800.     A  third  person  put  out  $15,000  more  than  the  first,  at  a 
rate  per  cent,  greater  by  2,  and  received  an  income  greater  by 
$1 ,500.     Required  the  three  principals,  and  the  respective  rates  of 
interest. 

NOTE  2.  To  avoid  the  inconvenience  of  large  numbers  in  the  operation, 
put  a  =  5000;  then  2a  -  10COO,  3«  =  15000,  ^-  =  1500,  and  |^  =  800. 
In  the  final  result,  the  value  of  a  may  be  restored. 

Ans     j  Principals,  $30,000.  $40,000,  $45,000. 

(  Hates,  4,  5,  6,     per  cent. 

11 


122  SIMPLE    EQUATIONS. 

19.  If  B's  age  be  subtracted  from  A's,  the   difference  will  be  C's 
nge ;  if  5  times  B's  age  and  twice  C's  age  be   added   together,  and 
from  their  sum  A's  age  be  subtracted,   the   remainder  will  be  147; 
and  the  sum  of  the  three  ages  is  96.      Required  the  ages  of  A,  B, 
and  C,  respectively.  Ans.  A's,  48  ;  B's,  33  ;  C's,  15. 

20.  Find   what  each   of   three  persons,  A,  B,  and  C,  is  worth, 
knowing,  1st,  that  what  A  is  worth  added  to  3  times  what  B  and  C 
are  worth,  is  equal  to  4700  dollars;  2d,  that  what  B  is  worth  added 
to  4  times  what  A  and  C   are  worth,  is  equal   to   5800  dollars ;  3d, 
that  what  C  is  worth  added  to  5  times  what  A   and  B  are  worth,  is 
equal  to  6300  dollars.  Ans.  A,  $500  ;  B,  $600  ;  C,  $800. 

21.  A  grocer  sold  50  pounds  of  tea  at  an  advance  of  10  per  cent, 
on  the  cost,  and  30  pounds  of   coffee  at  an  advance  of  20  per  cent, 
on   the   cost,  and   received  for  the  whole  $27.40,  gaining  $2.90. 
What  was  the  cost  per  pound  of  the  tea  and  coffee  ? 

Ans.  Tea,  $.40;  Coffee,  $.15. 

22.  Five  persons,  A,  B,  C,  D,  E,  play  at  cards  ;  after  A  has 
won  one  half  of  B's  money,  B  one  third  of  C's,  C  one  fourth  of  D's, 
D  one  sixth  of  E's,  they  have  each  $30.     How  much  had  each  to 
begin  with?      Ans.  A,  $11 ;  B,  $38  ;  C,  $33  ;  D,  $32  ;  E,  $36. 

23.  Three  brothers  desired  to  make  a  purchase,  requiring  $2000 
of  each.     The  first  wanted,  in  addition  to  his  own  money,  4  of  the 
money  of  the  second ;  the  second  wanted,  in  addition  to  his  own, 
|  of  the  money  of  the  third ;  and  the  third  wanted,  in  addition  to 
his  own,  I  of  the  money  of  the  first.     How  much  money  had  each  ? 

Ans.  1st,  $1280;  2d,  $1440;  3d,  $1680. 

9 

24.  A  courier  was  sent  from  A  to  B,  a  distance  of  147  miles; 
after  28   hours  had  elapsed,   a  second  courier  was  sent  from  the 
same  place,  who  overtook  the  first  just  as  he  entered  B.     Now  the 
time  required  by  the  first  to  travel   17  miles,  added  to  the  time  re- 
quired by  the  second  to  travel  56  miles,  is  13|  hours.     How  many 
miles  did  each  travel  per  hour?      Ans.  1st,  3  miles;  ?d,  7  miles. 

25.  Find  two  numbers,  such  that  if  J  of  the  greater  be  added  to 
J  of  the  less,  the  sum  shall  be  13  ;  and  if  4  of  the  less  be  subtract- 
ed  from  |  of  the  greater,  the  remainder  will  be  nothing. 

Am.  18  and  12. 


TWO    OR    MORE    U1T KNOWN    QUANTITIES.  123 

26.  Find  three  numbers  of  such  magnitudes,  that  the  first  added 
to  \  of  the  sum  of  the  other  two,  the  second  added  to  ^  of  the  sum 
of  the  other  two,  and  the  third  added  to  ]  of  the  sum  of  the  other 
two,  may  each  be  equal  to  51.  Ans.  15,  33,  and  39. 

27.  Said  A  to  B  and  C,  "  If  each  of  you  wifl  give  me  4  sheep, 

I  shall  have  4  more  than  both  of  you  will  have  left."     Said  B  to  A 
and  C,  "  If  each  of  you  will  give  me  4  sheep,  I  shall  have  twice  as 
many  as  both  of  you  will  have  left."     Said  C  to  A  and  B,  "  If  each 
of  you  will  give  me  4  sheep,  I  shall  have  three  times  as  many  as 
both  of  you  will  have  left."     How  many  sheep  had  each  ? 

Ans.  A,  6 ;  B,  8 ;  C,  10. 

28.  "What  fraction  is  that,  to  the  numerator  of  which  if  1  be 
added,  the  fraction  will  be  ^  ;  but  if  to  the  denominator  1  be  added, 
the  fraction  will  be  |  ?  Ans.  T4-. 

29.  What  fraction  is  that,  to  the  numerator  of  which  if  2  be 
added,  the  fraction  will  be  |  j  but  if  to  the  denominator  2  be  added, 
the  fraction  will  be  -g  ?  Ans.  ^. 

30.  Four  persons,  A,  B,  C,  D,  were  engaged  together  in  mowing 
for  4  successive  days.     The  first  day  A  worked  1  hour,  B  3  hours, 
C  2  hours,  and  D  2  hours,  and  all  together  mowed  1  acre  ;  the 
second  day  A  worked  3  hours,  B    2  hours,  0  4  hours,   and  D   11 
hours,  and  all  together  mowed  2  acres  ;  the  third  day  A  worked  5 
hours,  B  4  hours,  C  12   hours,  and  D   5  hours,  and  all  together 
mowed  3  acres  ;  the  fourth  day  A  worked  9  hours,  B  7  hours,  C  6 
hours,  and  D  8  hours,  and  all  together  mowed  4  acres.     How  many 
hours  would  each  alone  require  to  mow  1  acre  ? 

Ans.  A,  5  hours ;  B,  6  hours  ;  C,  12  hours ;  D,  15  hours. 

31.  If  A  give  B  $5  of  his  money,  B  will  have  twice  as  much 
money  as  A  has  left ;  and  if  B  give  A  $5,  A  will  have  thrice  as 
much  as  B  has  left.     How  much  has  each  f 

Ans.  A,  $13;  B,  $11. 

32.  A  corn  factor  mixes  wheat  flour,  which  cost  him  10  shillings 
per  bushel,  with  barley  flour,  which  cost  4  shillings  per  bushel,  in 
such  proportion  as  to  gain  43|  per  cent,  by  selling  the  mixture  at 

II  shillings  per  bushel.     Required  the  proportion. 

Ans.  The  proportion  is  14  bu?hels  of  wheat  flour  to  9  of  barley. 


124  SIMPLE    EQUATIONS. 

33.  There  is   a  number  consisting  of  two  digits,  which  number 
divided  by  5  gives  a  certain  quotient  and  a  remainder  of  1,  and  the 
same  number  divided  by  8  gives  another  quotient  and  a  remainder 
of  1.     Now  the  quotient  obtained  by  dividing  by  5  is  twice  the  val- 
ue of  the  digit  in  the  tens'  place,  and  the  quotient  obtained  by  divi- 
ding by  8  is  equal  to  5  times  the  digit  in  the  units'  place.     What  is 
the  number  ?  Ans.  41. 

34.  The  four  classes  in  a  certain  college  are  to  compete  for  four 
prizes,  amounting  in  the  aggregate  to  $119,  and  the  prize  money  is 
to  be  raised  by  contribution,  on  the  following  conditions,  namely : 
that  the  members  of  the  class  whose  candidate  obtains  the  1st  prize 
shall  each  pay  one  dollar,  and  the  class  whose  candidate  obtains  the 
2d  prize  shall  pay  the  remainder.     Now  it  is  found  that  if  a  senior 
gets  the  1st  prize  and  a  junior  the  2d,  each  junior  will  pay  ^  of  a 
dollar ;  if  a  junior  gets  the  1st  prize  and  a  sophomore  the  2d,  each 
sophomore  will  pay  \  of  a  dollar ;  if   a  sophomore  gets  the  1st  prize 
and  a  freshman  the  2d,   each  freshman   will  pay  ^  of  a  dollar ;  and 
if  a  freshman  gets  the   1st  prize  and  a  senior  the  2d,  each  senior 
will  pay  |  of  a  dollar.      Of   how  many  members  does  each  class 
consist  ?  A        j  Freshman,  104 ;  Sophomore,  9o ; 

(  Junior,         88 ;  Senior,          75. 

35.  Find  four  numbers,  such  that  if  3  times  the  first  be  added  to 
the  second,  4  times  the   second  be   added  to   the  third,  5  times  the 
third  be  added  to  the  fourth,  and  6  times  the  fourth  be  added  to  the 
first,  each  sum  shall  be  359.  Ans.  95,  74,  63,  44. 


GENERAL  SOLUTION  OF  PROBLEMS. 

fi  7o.  In  the  preceding  problems,  the  given  quantities  have  been 
expressed  by  numbers,  and  it  has  been  required  simply  to  determine 
the  values  of  the  unknown  quantities  from  the  numerical  relations 
thus  expressed. 

Tf,  however,  the  given  quantities  in  any  problem  be  represented 
by  letters,  the  solution  will  give  rise  to  a  formula,  showing  not  only 
the  value  of  the  unknown  quantity,  but  indicating  the  precise  ope- 


GENERAL   PROBLEMS.  123 

rations  to  be  performed  in  order  to  obtain  this  value.     This  is  called 
a  general  solution  of  the  problem. 

176.  When  any  particular  problem  has  been  proposed,  we  may, 
by  simply  varying  the  numbers,  form  other  problems  of   the  same 
kind  or  class ;  and  the  solutions  of  all  the  problems  of  the  class  will 
require  exactly  the  same  operations.     Hence, 

177.  The  General  Solution  of   a   problem   is  the  process  of 
obtaining  a  formula  which  shall  express,  in  known  terms,  the  values 
of  the  unknown  quantities  in  the  given  problem,  or  in  any  problem 
of  its  class. 

178.  An  Arbitrary  Quantity  is  one  to  which  any  value  may  be 
assigned  at  pleasure,  in  a  general  formula  or  equation. 

179.  For  illustration,  let  the  following  questions  be  proposed  : 

1  — What  number  is  that  whose  third  part  exceeds  its  fourth  part 
by  6? 

Instead  of  confining  our  attention  to  the  particular  numbers  here 
given,  we  may  first  investigate  the  problem  under  a  general  form,  as 
follows : 

What  number  is  that  whose  mth  part  exceeds  its  nth  part  Ly  a  ? 

Let  x  represent  the  number ;  then  by  the  conditions, 

(1) 

clearing  of  fractions,          nx — mx  =  amn,  (2) 

amn 

whence,  x  = (3) 

n — m 

Equation  (3)  is  the  formula  which  indicates  the  operations  to  be 
performed  in  solving  all  questions  of  this  class. 

If  in  this  formula  we  put  m  =  3,  n  =  4,   and  a  =  6,  we  shall 


4—3 

the  number  required  by  the  particular  question  as  at  first  proposed. 

2. — What  number  is  that  whose  fifth  part  exceeds  its  seventh 
part  by  12  ? 

To  obtain  the  number  by  the  formula,  let  m  =  5,  n  =  7,  and 
a  =  12;  then 

12x5x7 

x  =  — - — = —  =  210,  Arts. 
7 — o 


126  SIMPLE    EQUATIONS. 


EXAMPLES  FOR  PRACTICE. 

1.  Divide  the  number  n  into  two  such  parts  that  the 
increased  by  a  shall  be  equal  to  the  less  increased  by  b. 

n-f-'£ — a                  n-\~a — b 
Ans.  Greater, — —  :     Less,  -      5 

4  u 

2.  In  the  last  example,  what  will  be  the  two  parts  if  n  =  84, 
a  =  16,  and  b  =  58  ?  Ans.  63  and  21. 

3.  The  sum  of  three  numbers  is  s ;    the  second  exceeds  the  first 
by  a,  and  the  third  exceeds  the  second  by  b.     Required  the  numbers. 

s — 2a — b     s-f-a — b     a-fa+26 


4.  My  indebtedness  to  three  persons,  A,  B,  and  C,  amounts  to 
a  dollars ;  and  I  owe  B  n  times  the  sum  which  I  owe  A,  and  C  m 
times  the  sum  which  I  owe  A.     What  is  my  indebtedness  to  A  ? 

Ans.   -iii* 

5.  In  the  last  example,  what  is  the  sum  due  to  A  when  a  =  $786, 
n  =  2,  and  m  =  3  ?  Ans.  $131. 

6.  A  person  engaged  to  work  a  days  on  these  conditions :  For 
each  day  he  worked  he  was  to  receive  b  cents,  and  for  each  day  he 
was  idle  ^e  was  to  forfeit  c  cents ;  at  the  end  of  a  days  he  received 
d  cents.     How  many  days  was  he  idle  ?  ab — d  , 

b+c 

7.  My  horse   and  saddle  are  together  worth  a  dollars,  and  my 
horse  is  worth  n  times  the  price  of  my  saddle.      What  is  the  value 

°f  Cach?       .  A».  Saddle,  -^L  j  Horse,    ~ 


8.  The  rent  of  an  estate  is  n  per  cent,  greater  this  year  than 
it  was  last.     This  year  it  is  a  dollars  ;  what  was  it  last  year  ? 

lOOa 

Ans-          d 


9.  A  person  after  spending  a  dollars  more  than  J  of  his  income, 
*iad  remaining  b  dollars  more  than  ^  of  it.     Required  his  income. 


Ans.  doUars. 


GENERAL    PROBLEMS.  127 

10.  A  person  after  spending  a  dollars  more  than  ^th    of  his   in- 
come, had  remaining  b  dollars  more  than  -^th  of  it.     Required  his 


ncome- 


dollars. 


mn  —  in  —  n 

11.  If  A  can  perform  a  certain  piece  of  worfcf  in  a  days,  and  B 
can  do  the  same  in  b  days,  and  C  the  same  in  c  days,  in  how  many 
days  can  all  together  perform  the  work  ? 

abc 

Ans.  —j—.  —  7-7—  days. 
ab-\-ac-\-bc 

12.  In  the  last  example,  what  will  be  the  time  required,  when 
a  =  6,  b  =  8,  and  c  =  12  ?  Ans.  2f  days. 

13.  If  from  a  times  a  certain  number  c  be  subtracted,  the  remain- 
der will  be  equal  to  b  times  the  number  increased  by  d.     llequired 

the  number.  A        c-\-d 

Anx.   —  !  __ 

a—  b 

14.  A  farmer  would  mix  oats  worth  a  cents  a  bushel  with  peas 
worth  b  cents  a  bushel,  to  form  a  mixture  of  c  bushels  worth  d 
cents  a  bushel.     How  many  bushels  of  each  kind  must  he  take  ? 

Ans.    Oats,    1^;  Peu,     fc9. 
a  —  b  a  —  b 

15.  There  were  a  boys  in  one  party,  and  b  boys  in  another  party, 
and   each  party  had   the  same  number  of  nuts.     Each  boy  in  the 
first  party  snatched  m  nuts  from  the  second  party,  and  ate  them  ; 
then  each  boy  in  the  second  party  snatched  m  nuts  from  the  first 
party,  and  ate  them.     Each  party  then  divided  the  nuts  remaining 
to  it  equally  among  its  members,  when  the  boys  in  the  two  parties 
found  that  they  had  the  same  number  of  nuts  apiece  ;  how  many 
nuts  had  each  party  at  first  ?  Ans.  r/t(a-j-£/). 

16.  Find  four  numbers,  such  that  if  a  times  the  first  be  added 
to-the  second,  b  times  the  second  be  added  to  the  third,  c  times  the 
third  be  added  to  the  fourth,  and  d  times  the  fourth  be  added  to 
the  first,  each  sum  shall  be  m. 


iit(b(  d — cd-\-d — 1)  m(acd — ad-\-a — 1" 

1st. r~7~~i >  2d'  nr~i — ' 

i  aueu — 1  liocd — I 

m(abd — ab-{-b — 1)  m(abc — bc-\-c — 1} 

3d,    — - — =• :  4th,  ^—-7 — r 

abed — 1  abed — 1 


128  SIMPLE   EQUATIONS. 

17.  A  sent  n  pupils  regularly  to  a  certain  school  during  a  term 
of  a  days,  and  B  sent  m  pupils  regularly  to  another  school  for  a 
term  of  I  days.  The  two  schools  had  the  same  number  of  pupils 
in  attendance,  and  raised  the  same  amount  of  money  by  rate-bill. 
There  were  c  days'  absence  allowed  for  at  the  school  to  which  A 
sent,  and  d  days'  absence  at  the  school  to  which  B  sent ;  and  A 
and.B  found  that  they  had  equal  sums  to  pay.  What  was  the 

number  of  pupils  attending  each  school  ?  bcm — arfn 

Ans.  —r-. 7— 


18.  Divide  the  number  m  into  four  parts,  such  that  the  second 
shall  be  a  times  the  first,  the  third  a  times  the  second,  and  tho 

fourth  a  times  the  third.  in 

An*.  1st  part, 


19.  The  sum  of   two  numbers  is  s,  and  their  difference  is  d. 

Required  the  numbers.  s-\-d  s  —  d 

Ans.    Greater,  —  -—  ;    Less,    —  •=  — 

J  ^ 

20.  There  are  three  numbers,  such  that  the  sum  of  the  first  and 
second  is  a,  the  sum  of  the  first  and  third  is  I,  and  the  sum  of  the 
second  and  third  is  c.     What  are  the  numbers  ? 

a-\-b  —  c          a-4-r  —  b    „  ,    l-\-c  —  a 
Ans.  1st,  -^2  —  ;  2d,  -^—  ',  3d;  -3L  -- 

21.  There  is  a  number  consisting  of  two  digits  ;  the  number  is 
equiil  to  a  times  the  sum  of  its  digits  ;  and  if  c  be  added  to  the 
number',  the  order  of  the  digits  will  be  reversed.     Required  tho 

two  digits.  t.(10_a) 

V   Digit  in  units'  place,  -—-—  -; 

Ans.    *  c(a—  n 

Digit  in  tens'  place, 


22.  Find  what  each  of  three  persons,  A,  B,  and  C,  is  worth, 
knowing,  1st,  that  what  A  is  worth  added  to  I  times  what  B  and 
C.  arc  worth  is  equal  to  p  ',  2d,  that  what  B  is  worth  added  to  m 
times  what  A  and  C  are  worth  is  equal  to  q  ;  3d,  that  what  C  is 
worth  added  to  n  times  what  A  and  B  are  worth  is  equal  to  r. 

We  give  here  a  solution  of  this  example,  partly  to  illustrate  tho 
method  of  simplifying  algebraic  formulas  by  the  use  of  auxiliary 
quantities. 


GENERAL    PROBLEMS.  129 

Let  x  =  A's  money,  y  ==  B's  money,  z  =  C's  money. 


=  p>  (!) 

Then,  by  the  conditions,    4  y-j-mx-j-mz  =  ^,  (2) 

(  z-\-nx  -\-iiy  =  r.  (3) 

Assume  x+y-\-z  =  *r  (4) 

Multiplying  (4)  by  Z,  wi,  and  n,  sue-  /  x  =  - — -  j  (5) 
cessively,  and  subtracting  (1)  from  the   1 

first  product,  (2)  from  the  second,  and  /   y  = y  j  (6) 

(3)  from  the  third,  and  reducing  the    J 

respective  remainders,  we  have  f    z  =  —   -=•>  (7) 

N 

Adding  (5),  (6),  and  (7)3  we  obtain 

^-f-^T'  (8) 
i — 1    '    n — 1 


S  ~  (l—l  +  m-1  +  n—l)  "~  (l—l  +  m— 1  +  n— 1  j  (9) 

Now   the   parenthetical   expressions  in   equation  (0)   are   known 
quantities.     Hence,  to  simplify  the  results, 

I  m  .n 

-T  +  ;  ~T  +  — T'  (10) 

.  — 1  771 1          n L 

Put  < 

\  m  ft  ** 

(11) 


m_l    '   n_  1 

Equation  (?)  then  becomes     s=as  —  6,  (12) 

b 


whence  s  = 


o—l 


Substituting  value  of  *  in  (5),  x    =  ^^  —  =-^-  -  ^-  > 

(I  —  \.)(a  —  L) 


a—  1). 


"   (7),   «  = 


130  SIMPLE    EQUATIONS. 

DISCUSSION  OF  PROBLEMS  INVOLVING  SIMPLE  EQUATIONS 

180.  The  Discussion  of  a  problem  consists  in  attributing  certain 
values  and  relations  to  the  arbitrary  quantities  which  enter  the  equa- 
tion, and  in  interpreting  the  results. 

181.  When  a  problem  has  been  solved  in  a  general  manner,  we 
may  proceed  to  make  an  unlimited  number  of  suppositions  upon  the 
arbitrary  quantities  involved  in  the  formulas,  and  thus  obtain  a  va- 
riety of  results.     But  our  experience  of  algebraic  equations  would 
lead  us  to  expect  that  the  problem  might  not  be  rational,  or  possible, 
under  every  hypothesis.     Now  the  principal  object  in  the  discussion 
of  a  problem  is  to  examine  the  peculiar  or  anomalous  forms  which 
present  themselves,  and  ascertain  whether  the  problem  is  rational  or 
absurd,  or  how  it  is  to  be  understood,  under  the  suppositions  which 
lead  to  these  peculiarities.     We  shall  commence  with  the 

INTERPRETATION    OF    NEGATIVE   RESULTS. 

1.  What  number  must  be  added  to  a  that  the  sum  may  be  b  ? 
Let  x  represent  the  required  number.     Then  by  the  conditions  of 
the  question, 

a-fz  =  1}  (1) 

x  =  I— a.  (2) 

This  is  a  general  solution,  a  and  b  being  arbitrary  quantities. 

First,  suppose  a  =  20  and  b  =  28;  then  by  the  formula, 

x  =  28—20  =  8, 

a  result  which  satisfies  the  conditions  ;  for,  we  perceive  that  8  is  the 
number  which  must  be  added  to  20,  or  a,  to  make  28,  or  b.  Second^ 
suppose  a  =  20  and  b  =  12  ;  then  by  the  formula, 

x  =  12—  20  =  —  8, 
a  negative  result. 

In  order  to  ascertain  the  meaning  of  the  minus  sign  in  this  case, 
let  us  enunciate  the  question  according  to  the  supposition  that  gave 
this  result ;  thu-, 

What  number  must  be  added  to  20,  that  the  sum  may  be  12  ? 


DISCUSSION    OF    PROBLEMS.  131 

Now  as  20  is  greater  than  1 2,  no  number  can  be  added  to  20, 
arithmetically,  to  make  12.  The  problem  is  therefore  impossible 
under  the  second  hypothesis,  if  understood  in  an  arithmetical  sense. 

We  shall  find,  however,  that  if  we  change  the  words  added  to, 
and  sum,  to  their  opposites,  the  result  will  be  a  rational  question,  of 
which  8,  the  absolute  value  of  x,  is  the  answer.*  Thus, 

What  number  must  be  subtracted  from  20,  that  the  difference 
may  be  12  ?  Ans.  8. 

We  observe,  moreover,  that  the  negative  result,  — 8,  will  satisfy 
the  equation  of  the  problem,  under  the  second  hypothesis.  Thus, 

20+(-8J=12; 

or,  20—8  =  12. 

That  is,  12  is  really  the  algebraic  sum  of  20  and  — 8. 

2.  A  man  dying  left  two  sons,  the  elder  of  whom  was  a  years  of 
age,  and  the  younger  b  years  of  age.  In  how  many  years  after  the 
death  of  the  father  was  the  elder  son  twice  as  old  as  the  younger 
son? 

Let  x  represent  the  number  of  years ;  then  by  the  conditions, 

a+x  =  2(6+*)  ;  (1) 

x  =  a— 26.  (2) 

Since  a  and  b  are  arbitrary  quantities,  suppose  a  =  30  and 
b  =  12.  Then  by  the  formula, 

x  =  30—24  =  6. 

This  result  will  satisfy  the  conditions  arithmetically ;  for,  if  the 
elder  son  was  30,  and  the  younger  son  12  years  old,  at  the  death  of 
the  father,  then  in  6  years  the  age  of  the  elder  was  30-f-6  =  36 
years,  and  the  age  of  the  younger  was  12-J-6  =  18  years. 

Again,  suppose  a  =  30  and  b  =  18.     Then  by  the  formula, 
x  =  30—36  =  —6. 

To  interpret  the  negative  result  in  this  case,  we  observe  that  the 
problem  under  the  second  hypothesis  is  impossible,  if  understood 
iu  the  exact  sense  of  the  enunciation.  For,  when  the  elder  son 
was  30  and  the  younger  sen  18  years  old,  the  younger  son  was 
already  more  than  one  half  as  old  as  the  elder  ;  and  as  their  ages  are 
equally  increased  by  any  lapse  of  time,  it  is  evident  that  the  elder 


132  SU1PLK    EQUATIONS. 

son  could  never  become  twice  as  old  as  the  younger  son,  after  tho 
death  of  the  father.  Let  us  therefore  modify  the  general  problem 
as  follows  : 

A  man  dying  left  two  sons,  the  elder  of  which  was  a  years  of  age, 
and  the  younger  b  years  of  age.  How  many  years  before  the  death 
of  the  father  was  the  elder  son  twice  as  old  as  the  younger? 

If  we  let  x  represent  the  number  of  years,  then  the  solution  will 
be  as  follows : 

a—x  =  2(b— x);  (1) 

x  =  21— a.  (2) 

Now  suppose,  as  before,  that  a  =  30  and  I  =  18.  Then  by  tho 
new  formula, 

x  =  86—30  =  G, 

a  result  which  will  satisfy  the  modified  conditions;  for,  six  years 
before  the  death  of  the  father,  the  age  of  the  elder  was  30 — 6  =  24, 
and  the  age  of  the  younger  was  18 — 6  =  12. 

From  the  foregoing  discussions  we  draw  the  following  inferences  : 

1. —  When  the  solution  of  a  problem  by  a  simple  equation  gives  a 
negative  result,  the  minus  sign  indicates  that  the  problem  is  impossible, 
if  understood  in  the  exact  sense  of  the  enunciation. 

2. —  The  impossibility  thus  indicated  consists  in  adding  a  quantity 
when  it  should  be  subtracted  ;  or  in  treating  a  quantity  as  reckoned 
or  applied  in  a  certain  direction,  when  it  should  be  reckoned  or  ap- 
plied in  an  opposite  direction. 

3. — In  all  such  cases,  an  analogous  problem  may  lie  formed,  in- 
volving no  impossibility,  by  changing  the  terms  of  the  absurd  condi- 
tion to  their  opposite*;  and  the  answer  to  the  new  question  icill  be 
found  by  simply  changing  the  sign  of  the  negative  result  already 
obtained. 

18S.  The  foregoing  discussions  give  a  more  extensive  significa- 
tion to  the  plus  and  minus  signs,  and  lead  to  a  more  general  view  o£ 
positive  and  negative  quantities,  than  was  presented  in  a  former  sec- 
tion. 

Let  us  recur  to  the  problem  of  the  two  sons.  In  the  solution  of 
this  problem,  wo  employ  the  signs,  -f-  and  — ,  in  the  statement,  mere- 


NEGATIVE    RESULTS.  133 

ly  to  indicate  addition  and  subtraction.  But  in  the  result,  these 
signs  have  a  very  different  use ;  they  enable  us  to  distinguish  the 
circumstances  or  conditions  of  the  quantities  which  they  affect.  Thus, 
under  the  first  hypothesis,  the  period  of  time  represented  by  a-  oc- 
curred after  the  death  of  the  father,  and  in  the  result  is  found  to  be 
affected  by  the  plus  sign;  but  under  the  second  hypothesis,  the  pe-" 
riod  represented  by  x  occurred  before  the  death  of  the  father,  and 
in  the  result  is  found  to  be  affected  by  the  minus  sign. 

Thus  we  perceive  that  plus  and  minus,  in  Algebra,  are  not  symbols 
of  operation  merely,  but  also  symbols  of  relation,  serving  to  dis- 
tinguish quantities  in  opposite  conditions  or  circumstances. 

It  should  be  observed,  however,  that  this  enlarged  use  of  the 
plus  and  minus  signs  is  not  entirely  conventional  or  arbitrary,  but 
is  necessarily  involved  in  the  nfore  extended  signification  given  to 
the  terms  addition  and  subtraction,  in  Algebra.  Indeed  we  shall 
never  meet  with  a  negative  result  in  the  solution  of  problems,  so 
long  as  the  language  conforms,  in  the  exact  arithmetical  sense,  to 
the  facts  of  the  case. 


EXAMPLES   FOR   PRACTICE. 

1.  What  number  is  that  whose  fourth  part  exceeds  its  third  part 
by  12  ?  AM.  —144. 

The  question  is  impossible,  if  understood  in  an  arithmetical  sense. 
Let  the  pupil  modify  the  enunciation,  and  solve  the  new  problem. 

2.  A  man  when  he  was  married  was  30  years  old,  and  his  wife 
15.     How  many  years  must  elapse  before  his  age  will  be  three 
times  the  age  of  his  wife  ?  Ans.  — 7^  years. 

That  is,  their  ages  bore  the  specified  relation  7^  years  before,  not 
after,  their  marriage. 

3.  The  sum  of  two  numbers  is  s,  and  their  difference  d,  what  are 

the  numbers  ?  s        d  s        d 

Ans.    Greater,  —  -j-  «  >    Less,  ^-  —  — 

How  shall  the  result  be  interpreted  when  s  =  120  and  d  —  160  ? 

4.  Two  men,  A  and  B,  commenced  trade  at  the  same  time,  A 
liaving  3  times  as  much  money  as  B.     When  A  had  gained  $400 

12 


134  SIMPLE    EQUATIONS. 

and  B  $150,  A  had  twice  as  much  money  as  B  ;  how  much  did  each 
have  at  first?  Ans.  A  was  in  debt  $300,  and  B  $100. 

5.  A  man  worked  7  days,  and  had  his  son  with  him  3  days,  and 
received  for  wages  22  shillings,  and  the  board  of  his  son  and  him- 
self while  at  work.     He  afterward  worked  5  days,  and  had  his  son 
with  him  one  day,  and  received  18  shillings.     What  were  his  daily 
wages,  and  what  the  daily  wages  of  his  son  ? 

Ans.  The  father  received  4  shillings  per  day,  and  paid  2  shillings 
for  his  son's  board. 

6.  A  man  worked  for  a  person   10  days,  having  his  wife  with 
him  8  days,  and  his  son  6  days,  and  he  received  $10.30  as  compen- 
sation for  all  three ;  at  another  time  he  wrought  12  days,  his  wife 
10  days,  and  sou  4  days,  and  he  received  $13.20  ;  at  another  time 
he  wrought  15  days,  his  wife  10  days,  and  his  son  12  days,  at  the 
same   rates  as  before,  and  he  received  $13.85.     What  were  the 
daily  wages  of  each  ? 

Ans.  He  received  $.75  for  himself,  $.50  for  his  wife,  and  paid 
$.20  for  his  son's  board. 

7.  A  man  wrought  10  days  for  his  neighbor,  his  wife  4  days,  and 
eon  3  days,  and  received  $11.50  ;  at  another  time  he  served  9  days, 
his  wife  8  days,  and  his  son  6  days,  at  the  same  rates  as  before,  and 
receivecT  $12.00  ;  a  third  time  he  served  7  days,  his  wife  6  days, 
and   his  son  4   days,  at  the  same  rates  as  before,  and  he  received 
$9.00.     What  were  the  daily  wages  of  each  ? 

Ans.  Husband's  wages,"  $1.00;  Wife's,  0  ;  Son's,  $.50. 

8.  What  fraction  is  that  which  becomes  |  when   1  is  added  to 
its  numerator,  and  ^  when  1  is  added  to  its  denominator  ? 

Ans.  In  an  arithmetical  sense,  there  is  no  such  fraction.  The 
algebraic  expression,  zjf,  will  give  the  required  results. 

How  shall  the  enunciation  be  modified,  to  form  an  analogous 
question  involving  no  absurdity  ? 

9.  Four  merchants,  A,  B,  C,  D,  find  by  their  balance  sheets  that 
if  they  unite  in  a  firm,  receiving  the  assets  and  assuming  the  liabil- 
ities of  each,  they  will  have  a  joint  net  capital  of  $5780.     If  A,  B, 
and  C    unite  on  the  same  conditions,  their  joint  capital  will  be 
$7950 ;  if  B,  C,  and  D  unite,  their  joint  capital  will  be  $2220;  and 


NOTHING    AND    INFINITY. 

if  C,  D,  and  A  unite,  their  joint  capital  will  be   $7320.      Required 
the  net  capital  or  the  net  insolvency  of  each. 

10.  Two  men  were  traveling  on  the  same  road  towards  Boston,  A 
at  the  rate  of  a  miles  per  hour,  and  B    at  the  rate  of    b   miles   per 
hour.     At  6  o'clock  A  was  at  a  point  m  miles  f'rjpm  Boston,  and  at 
10  o'clock  B  was  at  a  point  n  miles  from  Boston.      Find  the  time 
when  A  passed  B  upon  the  road. 

A        m  —  n  —  4& 

Ans.  -     —  _  -  hours  after  6  o  clock. 
a  —  b 

11.  What  time  of  day  will  be  indicated  by  the  preceding  formu- 
la, if  m  =  36,  n  =  28,  a  =  5,  and  b  =  3  ?  Ans.  4  o'clock. 

12.  There  are  two  numbers  whose  difference  is  a;  and  if  3  times 
the  greater  be  added  to  5  times  the  less,  the  sum  will  be  b.     What 
are  the  numbers  ?  b--5a  b  —  3a 


Ans.  Greater.  —  5  —  ;  Less,  —  5- 
o  o 

How  shall  this  result  be  interpreted  if  a  =  24  and  b  =  48  ? 


NOTHING   AND   INFINITY. 

183.  The  limits  between  which  all  absolute  values  are  comprised, 
are  nothing  and  infinity  ;  and  the  symbols  by  which  these  limits  are 
denoted,  are  0  and  oo. 

184:.  In  certain  algebraic  investigations  it  is  convenient  to  em- 
ploy these  symbols  in  connection  with  each  other  and  the  ordinary 
symbols  of  quantity.  They  may  thus  sustain  the  relations  of  divi- 
sor, dividend,  quotient,  or  factor.  Such  relations,  however,  can 
not  really  exist  except  between  symbols  of  quantity.  Hence,  in 
Algebra,  0  does  not  always  signify  merely  absence  of  value  ;  nor  does 
oo  represent  infinity,  in  the  highest  sense  of  the  word. 

The  more  complete  definition  of  these  symbols  may  be  given  as 
follows  : 

185.  The  symbol  0,   called  nothing,   or  zero,  may  be  used  to 
denote  the  absence  of  value,  or  to  represent  a  quantity  less  than  any 
assignable  value. 

186.  The  symbol  GO,  called  infinity,  is  used  to  represent  a  quan- 
tity greater  than  any  assignable  value. 


136  SIMPLE    EQUATIONS. 


INTERPRETATION  OF  THE  FORMS    TT>    -  ,    ~     AND  -• 

0      GO     A1  0 

187.  In  order  to  understand  the  signification  of  the  expressions, 

A      A      U  0 

o'  -£  !'ando' 

we  may  consider  the  symbols  0  and  oo  as  resulting  from  an  arbitrary 
or  varying  quantity,  made  to  diminish  until  it  becomes  indefinitely 
small,  or  to  increase  until  it  becomes  indefinitely  great.  , 

a 

188.  Let    7     represent  a  fraction,  a  and  b  being  arbitrary  quan- 
tities.    And  let  it  be  remembered  that  the   value  of   a  fraction 
depends  simply  upon  the  relative    values  of   the  numerator  and 
denominator. 

1 — If  the  denominator  b  is  made  to  diminish,  becoming  less  and 
less  continually,  while  the  numerator  a  remains  unchanged,  the 
value  of  the  fraction  must  increase,  becoming  greater  and  greater 
continually,  (110,  II);  and  thus  when  the  denominator  b  becomes 
less  than  any  assignable  quantity,  or  0,  the  value  of  the  fraction 
must  become  greater  than  any  assignable  quantity,  or  OO.  Hence, 
we  conclude  that 

^  =  oo.  That  is, 

A  finite  quantity  divided  by  zero  is  an  expression  for  infinity. 

2.  — If  the  denominator  b  is  made  to  increase,  becoming  greatei 
and  greater  continually,  while  the  numerator  a  remains  unchanged, 
the  value  of  the  fraction  must  diminish,  becoming  less  and  less  con- 
tinually, (110.  II) ;  and  when  the  denominator  b  becomes  greater 
than  any  assignable  quantity,  or  oo,  the  value  of  the  fraction  must 
become  less  than  any  assignable  quantity,  or  0.  Hence, 

—  —  0  That  is, 

oo  ~ 

A  finite  quantity  divided  by  infinity  is  an  expression  for  zero  or 
nothing. 

3. --If  the  numerator  a  is  made  to  diminish,  becoming  less  and 
.ess  continually,  while  the  denominator  b  remains  unchanged,  the 


ANOMALOUS    FORMS,  181 

value  of  the  fraction  must  diminish  continually,  (119, 1)  j  and  when 
a  becomes  less  than  any  assignable  quantity,  or  0,  the  value  of  the 
fraction  also  must  become  0.  Hence, 

-  =  0-  That  is 

b 

Zero  divided  by  a  finite  quantity  is  an  expression  for  nothing  or 
zero. 

4. — If  both  a  and  b  are  made  to  diminish  simultaneously,  but  in 
such  a  manner  as  to  preserve  their  relative  value,  then  the  value  of 
the  fraction  will  remain  unchanged,  however  small  the  terms 
become,  (119,  III)  \  and  when  both  a  and  b  become  less  than  any 

assignable  quantity,  or  0,  we  shall  have  the  expression  ^-  reprcscnt- 

a  • 
ing  the  value  of  j-.     And  since  this  value  may  be  any  quantity 

0 
whatever,  we  conclude  that  ^-  represents  an  indeterminate  quantity. 

That  is, 

Zero  divided  by  zero  is  a  symbol  of  indetcrmination. 

NOTE. — If  it  should  be  difficult  for  any  one  to  conceive  how  both  terms 
of  a  fraction  may,  b}'  being  diminished,  become  nothing  at  the  same  time, 
and  yet  preserve  the  same  relative  value  to  the  last,  it  may  be  useful  to 
consider  the  following  illustrations : 
c 

Take  the  fraction  -5- ,  in  which  d  represents  the  diameter  of  a  circle, 

and  c  the  circumference.  Now  the  diameter  and  circumference  of  a  cir- 
cle have  the  same  ratio  to  each  other,  whatever  the  dimensions  of  the 
circle.  Hence,  if  the  circle  be  made  to  diminish  until  it  shall  become  a 

point,  or  vanish,  both  terms  of  the  fraction,  ^  ,  will  diminish,  and  become 

0  at  the  same  instant,  the  value  of  tJie  fraction  remaining  the  same  t.'irougJi- 

0 
out,  and  reducing  to  the  form,  Q->  at  the  instant  the  circle  vanishes.    Now 

the  ratio  of  the  diameter  to  the  circumference  of  a  circle  is  known  to  be 
£.1416—  ;  hence,  hi  the  present  case,  we  shall  have 

0 

O  =  8.1416. 

Again,  let  s  represent  the  side  of  a  square  and  d  the  diagonal.  Then 
we  have  the  well  known  ratio 

d_ 

If  the  square  is  supposed  to  diminish  by  insensible  degrees,  both  d  and 
9  will  vanish  at  the  same  instant,  and  we  shall  have  finally, 

0_ 

12* 


138  SIMPLE    EQUATIONS. 


PROBLEM  OF  THE  COURIERS. 

18O.  The  anomalous  forms  which  have  been  explained  in  the 
last  article  will  now  be  viewed  in  connection  with  a  general  problem, 
involving  certain  relations  of  motion,  time  and  distance.  The  dis- 
cussion will  also  confirm  our  interpretation  of  negative  results. 

PROBLEM. — Two  couriers,  A  and  B,  were  traveling  along  the 
same  road  and  in  the  same  direction,  namely,  from  C'  toward  C  ; 
the  former  going  at  the  rate  of  a  miles  per  hour,  and  the  latter  at 
the  rate  of  b  miles  per  hour.  At  12  o'clock,  A  was  at  a  certain 
point  P,  and  B  was  d  miles  in  advance  of  A,  in  the  direction  of  C. 
It  is  required  to  find  when  and  where  the  couriers  were  together. 


This  problem  is  entirely  general,  and  we  do  not  know  from  the 
enunciation  whether  the  couriers  were  together  after,  or  before  12 
o'clock ;  nor  whether  the  place  of  meeting  was  to  the  right,  or  to 
the  left  of  P.  But  in  order  to  effect  a  statement  of  the  problem, 
we  will  suppose  the  required  time  to  be  after  12  o'clock.  Then  we 
must  regard  time  after  12  o'clock  as  positive,  and  time  before  12 
o'clock  as  negative;  also,  distance  reckoned  from  P  toward  C  as 
positive,  and  distance  reckoned  from  P  toward  C'  as  negative.  Ac- 
cordingly, 

Let         t  =.  the  number  of  hours  after  12  o'clock  j 

x  =.  the  distance  from  P  to  the  point  of  meeting. 
And  since  A  traveled  at  the  rate  of  a  miles  per  hour,  and  B  at 
the  rate  of  I  miles  per  hour,  we  have 

x  •=.  at  =  distance  traveled  by  A  after  12  o'clock ; 

lit  =         "          "         "   B     t(       "         " 
But  since  A  and  B  were  d  miles  apart,  at  12  o'clock,  we  nave 
at— It  =  d, 

t =-*-.,  a) 

a — b 

x  =  ^L  (2) 

a—  b 

"We  may  now  discuss  this  problem  with  reference  to  the  time 
£,  and  the  distance  x,  which  are  the  two  unknown  elements 


PROBLEM    OF    THE    COURIERS.  139 

I.  Suppose  a>6. 

Under  this  hypothesis  the  values  of  both  t  and  x  will  be  positive, 
because  the  common  denominator,  a — b,  is  positive.  Now  since  t 
is  positive,  we  conclude  that  the  two  couriers  came  together  after 
12  o'clock  j  and  as  oc  is  positive,  we  infer  that  the  point  of  meeting 
is  somewhere  to  the  right  of  P. 

These  conclusions  agree  with  each  other,  and  are  consistent  with 
the  conditions  of  the  problem.  For,  the  supposition  that  a  is 
greater  than  b  implies  that  A  was  traveling  faster  than  B.  A 
would  therefore  gain  upon  B,  and  overtake  him  sometime  after  12 
o'clock,  and  at  a  point  situated  in  the  direction  of  C. 

II.  Suppose  a<6. 

Then  in  equations  (1)  and  (2)  the  denominator,  a — &,  is  negative, 
and  consequently  both  t  and  x  will  be  negative. 

This  implies  that  t  and  x  must  be  taken  in  a  sense  contrary  to 
to  that  in  which  they  were  employed  under  the  hypothesis,  (I), 
where  they  were  positive  ]  that  is,  the  time  when  the  couriers  were 
together  was  before  12  o'clock,  and  the  place  of  meeting  was  sit- 
uated to  the  left  of  P. 

This  interpretation,  also,  agrees  with  the  conditions  of  the  prob- 
lem, under  the  present  hypothesis.  For,  if  a  is  less  than  b}  then 
B  was  traveling  faster  than  A ;  and  as  B  was  in  advance  of  A  at 
12  o'clock,  he  must  have  passed  A  before  that  time,  somewhere  to 
the  left  of  P,  in  the  direction  of  C'. 

III.  Suppose  a  =  b. 

Under  this  hypothesis  we  shall  have  a—  b  =  0,  and 

d  ad 

t  3=  jT-  =  oo>   and  also   x  =  —=r-  =  oo. 

Now,  according  to  these  results,  t,  the  time  to  elapse  before  the 
couriers  are  together,  is  greater  than  any  assignable  quantity,  or 
infinity  ;  therefore  they  can  never  be  together.  And  likewise  x, 
the  distance  from  P  of  the  supposed  point  of  meeting,  is  greater 
than  any  assignable  quantity,  or  infinity  j  hence  there  can  be  no 
such  point,  however  distant  from  P. 

This  interpretation  is  in  accordance  with  the  conditions  of  the 
problem,  under  the  present  hypothesis.  For,  at  12  o'clock  the  two 


140  SIMPLE    EQUATIONS. 

couriers  were  d  miles  apart  ;  and  if  a  =  6,  they  were  traveling  at 
equal  rates,  neither  approaching  nor  separating.  Hence,  they  could 
always  continue  in  motion,  and  remove  to  any  distance  from  P, 
without  meeting. 

IV.  Suppose  d  =  0,  and  a>6  or  a<Jb. 
Then  we  shall  have 

=  -2-7=0,  and  a;=—  ^  =0. 
a  —  b 


That  is,  both  the  time  and  distance  are  nothing.  These  results 
must  be  interpreted  to  mean  that  the  couriers  were  together  at  12 
o'clock,  at  the  point  P,  and  at  no  other  time  or  place. 

And  this  interpretation  is  also  confirmed  by  the  conditions  of  the 
problem.  For,  if  d  =  0,  then  at  ]  2  o'clock  B  must  have  been  with 
A,  at  the  point  P.  And  if  a>6  or  a<&,  the  couriers  were  travel- 
ing at  different  rates,  and  must  be  either  approaching  or  receding 
from  each  other  at  all  times  except  at  the  moment  of  passing;  hence, 
they  could  be  together  only  at  a  single  point. 

Y.  Suppose  d  —  0,  and  a  =  b. 
We  shall  then  have 


Here  the  values  of  both  t  and  x  are  represented  by  the  symbol 
of  indetermination,  which  signifies  that  the  time  and  the  distance 
may  be  anything  whatever  ;  and  we  infer  that  the  couriers  must  be 
together  at  all  times,  and  at  any  distance  from  P. 

And  this  conclusion  is  evidently  confirmed  by  the  conditions  of 
the  problem.  For,  if  d  =  0,  the  couriers  were  together  at  12 
o'clock  ;  and  if  a  =  l>,  they  were  traveling  at  equal  rates,  and  would 
never  separate. 

1OO.  To   the    foregoing   interpretations,   there   is  an   apparent 

exception  in  the  case  of  the  expression   jr.     For,  a  fraction  which 
is  not  indeterminate  will  reduce  to  this  form,  if  its  terms  contain  a 
common  factor  that  becomes  zero  under  the  hypothesis. 
Thus,  in  the  solution  of  a  problem,  suppose 


DISCUSSIONS.  141 

If  we  put  a  =  b)  which  implies  that  a — b  =  0,  then 


and  the  value  of  x  appears  to  be  indeterminate.  Let  us,  however, 
cancel  the  common  factor,  a — ft,  from  both  terms  of  the  fraction  in 
equation  (1),  we  shall  obtain 

m-Q 


If  in  this  reduced  equation,  we  make  a  =  i,  as  before,  we  shall 
have  a  determinate'  value  for  x.  Thus, 

_3tt 

Hence  the  following  practical  direction  : 

» 
In  the  discussion  of  a  problem,    a  fractional  result  should  be 

reduced  to  its  lowest  terms  before  maJcing  the  hypothesis. 

1O1.  We  are  sometimes  liable  to  an  error  in  the  reduction  of  an 
equation,  in  consequence  of  a  false  assumption  respecting  the  valuo 
of  an  expression  reducible  to  the  form  of  indetermination. 

1.  —  Let  us  take  the  equation, 


x+2  ~      x—2 
Deducing  second  member,       --  —  =  6,  $) 

X-\-+j 

clearing  of  fractions,  6x-|-7  =  6z-f-12,  (3) 

transposing  and  factoring,    (6  —  6)x  =  5,  (4) 

55  ,~ 

whence,  x  =  ^-  -Q  =  g-, 

or,  by  (188,  1),  x=  co. 

This  result  is  erroneous.     To  obtain  the  true  root  of  equation  (1), 
multiply  both  members  by  (ai+2)  (x  —  2)  ;  we  shall  obtain 

6xa—  5,r—  14  =  6xa—  24  5 
whence.  5x  =  10, 

or,  x  =  2. 

Now  we  observe  that  if  this  true  value  of  x  be  substituted  in 
the  second  member  of  equation  (1),  it  will  reduce  to  the  form  ^r  > 


142  SIMPLE    EQUATIONS. 

and  the  mistake  in  our  first  solution  was  made  in   assuming  that 

6x—  12 

—  ^-  =  6,   a  conclusion  which  would  be  correct  in  all  cases 

except  when  x  =  2. 

2.—  If  we  make  two  assumptions  that  are  inconsistent,  respecting 
the  values  of  quantities  reducible  to  the  form  of  in  determination, 
the  result  will  be  an  algebraic  absurdity. 

Thus,  take  the  identical  equation, 

8+20  =  8+20.  (1) 

By  transposition,  8—8  =  20—20,  (2) 

8-8       20—20 
dividing  by  4  —  4,  T~\  ~  ~Z~f~' 


5(4—4) 
factoring,  ____  =  J, 


suppressing  common  factor,  and          2  =  5.  (5) 

Equation  (5)  is  absurd.  But  this  equation  is  not  correctly  derived 
from  (3)  or  (4).  In  equation  (3),  both  numerators  and  both  denomi- 
nators are  zero.  Hence  (3)  may  be  written, 

(L  __0. 
0  ~~  0' 

a  result  which  involves  no  absurdity,  and  certainly  gives  no  author- 
ity for  saying  that  2  is  equal  to  5. 

1O3.  To  afford  the  pupil  further  exercise  in  the  interpretation 
of  anomalous  forms,  we  give  the  following 

EXAMPLES. 

1 .  A  cistern  has  four  pipes  communicating  with  it.  If  all  be 
opened  together,  and  left  running  for  1 5  hours,  the  cistern  will  be 
filled  ;  but  if  the  first  run  only  5  hours,  the  second  8  hours,  the  third 
7  hours,  and  the  fourth  3  hours,  the  cisteru  will  be  but  one  half 
full ;  if  the  first  run  3  hours,  the  second  4  hours,  the  third  3  hours, 
and  the  fourth  1  hour,  only  -J  of  the  cistern  will  be  filled  ;  and  if  the 
first  run  4  hours,  the  second  2  hours,  the  third  3  hours,  and  the  fourth 
2  hours,  only  ]  of  the  cistern  will  be  filled.  In  what  time  would  the 
cistern  be  filled  by  each  pipe  alone  ? 


<S£t*^,^* 


DISCUSSIONS.  143 

2.  A  can  earn  5  dollars,  and  B  3  dollar?,  per  day.  In  2  days  A  will 
have  a  certain  sum,  and  in  4  days.B  will  have  2  dollars  more  than 
this  sum.  How  many  days  hence  will  A  and.B  have  the  same  sum?  6r6 

o.  An  astronomer  being  asked  the  period  of  a  comet's  revolution, 
answered,  that  if  from  3  times  the  period  10  ^ears  be  subtracted, 
and  to  4  times  the  period  8  years  be  added,  the  former  result  would 
be  equal  to  |  of  the  latter.  Required  the  period.  Xv/^vt-x^^TT" 

4.  Two  teachers,  A  and  B,  have  the  same  monthly  wages.  A  is 
employed  9  months  in  the  year,  and  his  annual  expenses  are  $450; 
B  is  employed  6  months  in  the  year,  and  his  annual  expenses  are 
$300.  Now  A  lays  up  in  two  years  as  much  as  B  does  in  3  years. 
Required  the  monthly  wages  of  each.  Q 


FORMULA  FOR  TIME  APPLIED  TO  CIRCULAR  MOTION. 

.  The  Problem  of  the  Couriers  gave  us  the  formula, 


in  which  a  and  I  are  the  respective  rates  of  motion,  d  the  distance 
to  be  gained,  and  t  the  time  to  elapse  before  the  couriers  will  be 
together. 

nut  the  relations  of  these  quantities  will  not  be  changed,  if  we 
suppose  the  path  of  motion  to  be  a  curve,  instead  of  a  straight  line. 
The  above  formula  will  therefore  apply  to  the  hands  of  a  clock 
moving  around  the  dial-plate,  or  to  the  planets  moving  in  the  circle 
of  the  heavens.  It  will  thus  afford  a  direct  solution  to  the  follow- 
ing problems  : 

1.  The  hour  and  minute  hands  of  a  clock  are  together  at  12 
o'clock  ;  when  are  they  next  together  f 

The  circumference  of  the  dial-plate  is  divided  into  12  spaces. 
The  minute  hand  moves  over  these  12  spaces  while  the  hour  hand 
moves  over  one  of  them  ;  and  when  the  minute  hand  has  gained 
upon  the  hour  hand  a  whole  circumference,  the  two  hands  will  be 
together. 

Taking  one  of  these  spaces  for  the  unit  of  distance,  and  one  hour 
for  the  unit  of  time,  we  have 


144  SIMPLE    EQUATIONS. 

.a  ='12,  6  =  1,  and  r7=:12, 
to  substitute  in  the  formula.     Hence, 

12          12 
«  =  5233=55=111.  5m.  27T\s. 

2.  At  what  time  between  2  and  3  o'clock  will  the  hour  and  minute 
hands  of  a  clock  be  together  ? 

In  this  dasc,  the  minute  hand  must  evidently  gain  two  revolutions, 
or  24  spaces.  Hence,  d  =  24  j  and  we  have  by  the  formula, 

t  =  ^  =  2h.  10m.  54T\s. 

3.  Wliat  time  between  2  and  3  o'clock  will  the  hour  and  minute 
liands  be  at  right-angles  to  each  other  f 

In  this  case  the  minute  hand  must  gain  21  revolutions ;  that  is, 
^=12X2^  =  27.  Hence, 

t  =  ^  =  2h.  27m.  lGT\s 

4.  What  time  between  5  and  6  o'clock  will  the  two  hands  of  % 
clc  Jc  be  in  the  same  straight  line  f 

Here  the  minute  hand  must  gain  5^  revolutions ;  and  d  =  12  X 
6£  =  66.  Hence, 

66 

f-ir»«- 

That  is,  the  hands  make  a  right  line  at  6  o'clock,  a  result  mani- 
festly true. 

We  will  now  apply  this  formula  to  certain  motions  of  the  heav- 
enly bodies.  It  is  known  that  the  moon  has  a  real  motion  around 
the  earth  from  west  to  east.  The  sun  also  has  an  apparent  motion 
in  the  same  direction,  in  consequence  of  the  real  motion  of  the 
earth  around  the  sun.  The  time  of  new  moon  is  when  the  moon  is 
ia  the  direction  of  the  sun  from  the  earth,  or  when  the  moon  is 
passing  the  sun,  in  her  motion.  With  this  explanation  we  present 
the  following  problem  : 

5.  The  average  daily  motion  of  the  moon  around  the  circle  of 
the  heavens  is  13.1764°,  and  the  apparent  daily  motion  of  the  sun 
in  the  same  direction  is  .98565°.     Required  the  time  from  one  new 
moon  to  another. 


INEQUALITT 


To  apply  the  formula,  we  have 
d  =  360°,  a  =  13.1764°,    b  =  .98565°,  and 
Hence, 


6.  The  planet  Venus,  as  seen  from  the  sun,  describes  an  arc  of 
1°  36'^per  day,  and  the  earth,  as  seen  from  the  same  point,  describes 
an  arc  q/*59'.  At  what  intervals  of  time  will  these  two  bodies  come 
in  a  line  with  the  sun  and  on  the  same  side  of  it? 

Hero  d  =  360°  =  21600',  a  =  1°  36',  and  b  =  59'.  Hence, 
a — b  =37',  and  we  have 

21600 
t  =  — |p—  =  583.8  days,  nearly. 

The  data  in  the  last  example  were  not  taken  with  extreme  accu- 
racy, the  object  being  mainly  to  illustrate  a  method.  More  exact 
data  would  have  given  583.92  days. 


INEQUALITIES. 

19*>.  An  Inequality  is  an  expression  signifying  that  one  quan- 
tity is  greater,  or  less,  than  another ;  as 

a>&,  and  c<r7. 

In  every  inequality,  the  part  on  the  left  of  the  sign  is  the  first 
member,  and  the  part  on  the  right  the  second  member. 

19$r.  In  treating  of  inequalities,  the  terms  greater  and  less, 
must  be  understood  in  their  algebraic  sense,  which  may  be  defined 
as  follows : 

Of  any  two  quantities,  as  a  and  b,  a  is  the  greater  when  a — b  is 
positive,  and  a  is  the  less  when  a — b  is  negative. 

19 7".  From  this  definition  it  follows,  that 

Any  negative  quantity  is  less  than  zero;  and  of  two  negative 
quantities,  the  greater  is  the  one  which  has  the  less  number  of  units. 

Thus, — 2<0,   because   — 2 — 0= — 2,   a  negative  result;    and 
— 3> — 5,  because  — 3 — ( — 5)  =  -f-2,  a  positive  result. 
13  K 


146  INEQUALITIES. 

198.  Two  inequalities  are  said  to  subsist  in  the  same  sense,  when 
the  first  member  is  the  greater  in  both,  or  the  less  in  both.  Thus 
a  >  d  and  c  >  d,  or  u  <  z  and  x  <  y,  are  inequalities  which  .sub- 
sist in  the  same  sense.  But  the  inequalities,  m  >  n  and  p  <  q, 
subsist  in  a  contrary  sense. 


PROPERTIES  OF  INEQUALITIES. 

199.  Inequalities  are  frequently  employed  in  mathematical  in- 
vestigations j  and  to  facilitate  their  use,  it  is  necessary  to  establish 
the  following  properties  : 

I.  An  inequality  will  continue  in  the  same  sense,  if  the  same  quan- 
tity be  added  to,  or  subtracted  from,  each  member. 

For,  suppose 

a  >  b. 

Then  according  to  (196),  a  —  b  is  positive.     Hence, 

(a±c}—(b±c) 
is  positive,  and  consequently 

a±c  >  b±c. 
It  follows  obviously  from  the  principle  just  established, 

1.  —  That  a  term  may  be  transposed  from   one  member  of   an  ine- 
quality to  another,  by  changing  its  sign. 

2.  —  That  if  an  equation  be  added   to  an  inequality,  member  to 
member,  or  subtracted  from  it  in   like  manner,  the  result  will  be  an 
inequality  subsisting  in  the  same  sense. 

II.  If  an  inequality  be  subtracted  from  an  equation,  member 
from  member,  the  sign,  of  inequality  -will  be  reversed. 

For,  suppose 

x=y,  and  a  >  5; 
then  we  shall  have  /)o  a.\  —  £  —  /y  —  fj  —  *. 


a  negative  quantity,  (196)  ;  hence, 

x  —  a  <^  y  —  b. 

III.  If  the  signs  of  all  the  terms  of  an  inequality  be  changed, 
the  sign  of  inequality  will  be  reversed. 


GENERAL    PRINCIPLES.  147 

For  to  change  the  signs  of  all  the  terms  is  equivalent  to  subtract- 
ing each  member  from  0=0. 

IV.  If  twu  or  more  inequalities  subsisting  '.n  the  same  sen.se,  be 
added,  member  to  member,  the  resulting  inequality  will  subsist  in  the 
same  sense  as  the  given  inequalities. 

For  if  a  >  b,  a'  >V,  a"  >  b", , 

then  from  (196), 

a—b,  a'—b',  a"—b", 

are  all  positive ;  and  the  sum  of  these  quantities, 

o_6_|_a'— 7/+o"— b",  or  (a-f  a'-f-a")— (&-f  &'+&"), 
is  therefore  positive.     Hence, 

a_j_a'_|_a"  >  6+64-6". 

It  is  evident  that  if  one  inequality  be  subtracted  from  another 
established  in  the  same  sense,  the  result  will  not  always  be  an 
inequality  subsisting  in  the  same  sense.  Thus,  it  is  evident  that 
we  may  have 

a  >  b  and  a1  >  b1, 

in  which  a — a'  may  be  greater  than  b — b' ,  less  than  b — b',  or  equal 
to  6—6'. 

V.  If  one  inequality  be  subtracted  from  another  established  in  a 
contrary  sense,  the  result  will  be  an  inequality  established  in   the 
same  sense  as  the  minuend. 

For,  if  a  >  b  (1) 

and  .  a'<6',  (2) 

then  a — 6  is  positive  and  a' — br  is  negative  ;  therefore,  a — 6 — 
(a' — 6'),  or  its  equal  (a — a') — (6 — b')  must  be  positive,  and  wo 
shall  have 

a— a'  >  6—6', 
an  inequality  subsisting  in  the  same  sense  as  (1). 

If  (1)  be  subtracted  from  (2),  member  from  member,  it  can  be 
shown,  in  like  manner,  that 

a'~ a  <  b'— 6. 

VI.  An  inequality  will  still  subsist  in  the  same  sense,  if  both 
members  be  multiplied  or  divided  by  the  same  positive  quantity. 


148  INEQUALITIES. 

For  suppose  m  to  be  essentially  positive,  and 

a>6. 

Then  since  a — b  is  positive,  we  shall  have  both  m(a — 6)  and 
—  (a — 6)  positive.  Therefore, 

ma  >>  nib  and  —  ">  —  • 

m       wi 

VII.  T/"  &o£&  members  of  an  inequality  be  multiplied  or  divided 
by  the  so.me  negative  quantity,  the  sign  of  inequality  will  be  reversed. 

For,  to  multiply  or  divide  by  a  negative  quantity  will  change  the 
signs  of  all  the  terms,  and  consequently  reverse  the  sign  of  inequal- 
ity, (III). 

VIII.  //  two  inequalities  subsisting  in  the  same  sense  be  multiplied 
together,  member  by  member,  the  sign  of  inequality  remains  the  same 
when  morf  than  two  of  the  members  are  positive,   but  is  reversed 
when  more  than  two  of  the  members  are  negative. 

That  is, 

Multiply       a  >  b         — a  >  — 6     — a  >  — b         a^>      b 
By  a'>b'       —  a'  >  —  b'      a'  >  —  V       a'  >  —  V 

Products,  aa'  >  bb'       aaf  <  bb'  —aa'  <      bb'  aa'  >  —  W 

The  first  two  results  are  evident  from  the  fact  that  when  the  two 
members  of  an  inequality  are  both  positive,  the  greater  member  has 
the  greatest  numerical  value ;  but  when  the  two  members  are  both 
negative,  the  greater  mem1  ,v  has  the  least  numerical  value. 

The  other  two  results  are  evident  from  the  fact  that  any  positive 
quantity  is  greater  than  any  negative  quantity. 

It  will  be  found  thai  if  two  of  the  four  members  arc  positive  and 
two  negative,  iLe  result  will  be  indefinite. 


REDUCTION   OP   INEQUALITIES. 

2OO.  The  Reduction  of  an  inequality  consists  in  transforming  it 
in  such  a  manner  that  one  member  shall  be  the  unknown  quantity 
standing  alone,  and  the  other  member  a  known  expression.  The 
inequality  will  then  denote  one  lim'U  of  the  unknown  quantity. 


REDUCTION.  149 

2O1.  The  principles  just  established  may  now  be  applied  in  the 
reduction  of  inequalities  of  the  first  degree. 

Thus,  let  it  be  required  to  find  the  limit  of  .t  in  the  inequality, 

x        2x      3x      9 

2  "T^T+r 
Multiplying  both  sides  by  20, 


transposing  and  collecting  terms, 

3z>45; 
dividing  by  3, 

x       15. 


EXAMPLES    FOB    TBACTICE. 


2x       2x       2x 
2.    ___>__2. 

6x       5       11        Ix 


3x       x—  1  20*-fl3 

4.  -j  ---  ^-  <  6x  --  j  —  •  ^Lns.  re  >  5. 

l+d 

5.  ax  —  6  >  cx-\-d.  Am.   x  >  -- 

a  —  e 

a:  —  a  x 

6.  —  =  —  <T  1  --  •  -4w«.  x  <  a. 

6  a 

7.  fa  —  a:^  Cwi  —  x}  —  a(m  —  c")  <^  cr*  --  •       jl?is.  a;^>  — 

m  »?i 

2O2.  If  there  be  given  an  inequality  and  an  equation,  contain- 
ing two  unknown  quantities,  the  limit  of  each  unknown  quantity 
may  be  found,  by  a  process  of  elimination. 

1.  Given  2x-\-5y  >  16  and  2x-\-y  =  12,  to  find  the  limits  of  x 
and  y. 

13* 


150  INEQUALITIES. 

If  we  subtract  the  equation  from  the  inequality,  the  result  will 
be  an  inequality  subsisting  in  the  same  sense,  (19O?  I,  2),  and  x 
will  be  eliminated.  Thus, 

From  2x+5y  >  16,  (1) 

subtract  2x+y  =  12 ;  (2) 


If  we  substitute  1  for  y  in  the  equation,  the  first  member  will  be 
made  less  than  the  second ;  and  we  shall  have 

2x+ 1  <  12, 
whence,  x  <  5^. 

The  limit  of  x  may  be  found  in  a  different  manner,  as  follows: 

From  equation  (2),  y  =  12 — 2x. 

Substituting  this  value  of  y  in  (1),  we  have 
2x-f60— 10*  >  16, 

whence,  — Sx  ]>  — 44, 

or,  x  <  5^. 

Thus  we  may  eliminate  between  equalities  and  inequalities,  either 
by  addition  and  subtraction,  or  by  substitution.  Let  it  be  remem- 
bered, however,  that  when  an  inequality  is  subtracted  from  an  equa- 
tion, the  sign  of  inequality  will  be  reversed;  (1O05  II). 

EXAMPLES  FOR  PRACTICE. 

1.  Given  2x-\- 4y  >  30  and  3x+  2y  =  31,  to  find  the  limits  of  a 
and  y.  Ans.  x  <  8  ;  y  >  3j.  ' 

2.  Given  4x— 3y  <  15  and  8x+2y  =  46,  to  find  the  limits  of  x 
and  y.  Ans.  x  <  5| ;  y •>  2. 

3.  Given  7* — Wy  <  59  and  4x-\- 5y  =  68,  to  find  the  limits  of  x 
andy.  Ans.  x  <  13  ;  y  >  31. 

4.  Given  5z-|-8y  >  121  and  7x-{-4y  =  168,  to  find  the  limits  of 
x  and  y.  Ans'.  x  <  20  ;  y  >  7. 

x— 4       ?/— 10  ,  3#— 24       a— y 

5.  Given  -g ^-g—  >  1   and  — ^ 1-  -^  =  13,  to 

find  the  limits  of  x  and  y.  Ans.  x  <  22f  ;  y  <  17f . 


POWERS   OF   MONOMIALS.  151 


SECTION    III. 

POWERS    AND    ROOTS. 
INVOLUTION. 

.  A  Power  of  a  quantity  is  the  product  obtained  by  taking 
the  quantity  some  number  of  times  as  a  factor ;  the  quantity  is  then 
said  to  be  raised,  or  involved. 

2O4.  Involution  is  the  process  of  raising  a  quantity  to  any 
given  power. 

2O«>.  Involution  is  always  indicated  by  an  exponent,  which 
expresses  the  name  of  the  power,  and  shows  how  many  times  the 
quantity  is  taken  as  a  factor. 

.Thus,  let  a  represent  any  quantity  whatever ;  then, 
The  first  power  of  a  is  a  =  a1 j 

"     second     "          "  aa  =  a* ; 

"     third       "          "  aaa  =  a*', 

"    fourth     "          "  aaaa  =  a4; 

"     nth          "         "  aaa...=an. 

2O6.  The  Square  of  a  quantity  is  its  second  power ;  and 
The  Cube  of  a  quantity  is  its  third  power. 

2O  7.  A  Perfect  Power  is  a  quantity  that  can  be  exactly  pro- 
duced by  taking  some  other  quantity  a  certain  number  of  times  as 
a  factor.  Thus,  x9 — Ixy+y*  is  a  perfect  power,  because  it  is  equal 
to  (x—y)  (x—y). 

POWERS    OF    MONOMIALS. 

2O8.  A  simple  factor  may  be  raised  to  any  power  by  giving  it 
an  exponent  which  expresses  the  name  or  degree  of  the  required 
power.  And  if  a  quantity  consists  of  two  or  more  factors,  it  is 
evident  that  as  often  as  the  quantity  is  repeated,  each  factor  will  be 
repeated.  Thus, 

a  =  aby  ab  =  aaXbb  =  a*b\ 


152  INVOLUTION. 

And  in  general,  if  abc  .....  k  represent  the  product  of  any 
number  of  factors,  and  n  any  exponent,  we  shall  have 

(ale  .....  A:)*1  rr=  ani*c»  .....  k*.  That  is, 

The  nth  power  of  the  product  of  two  or  more  factors  is  equal  to 
the  product  of  the  nth  powers  of  those  factors. 

2O9.  If  it  be  required  to  involve  a  quantity  which  is  already 
a  power,  the  exponent  of  the  quantity  will  be  taken  as  many  times 
as  there  are  units  in  the  exponent  of  the  required  power.     Thus, 
(««)*  =  amXam  =  0™+™  =  a*m  ; 
(a"1)3  =  amX«"*X"w  =  a"1*"1*"*  =  a3"*. 

And  in  general,  an  raised  to  the  nth  power  will  bo 

(am)n  =  amn.  That  ig, 

If  the  mth  power  of  a  quantity  be  raised  to  the  nth  power,  (lie 
result  will  be  a  power  of  the  quantity  expressed  by  the  product  of 
m  and  n. 

^1O.  "With  respect  to  signs,  it  is  obvious  that  if  a  positive  quan- 
tity be  involved  to  any  power  whatever,  the  result  will  be  positive. 

But  if  a  negative  quantity  be  involved,  the  successive  powers  will 
be  alternately  positive  and  negative  ;  for,  it  has  been  shown  that 
the  product  of  an  even  number  of  negative  factors  is  positive,  and 
the  product  of  an  odd  number  of  negative  factors  is  negative,  (G7). 

To  deduce  this  law  of  signs  in  an  experimental  way,  let  it  bo 
required  to  involve  —  a  to  successive  powers.  By  the  principles  of 
multiplication,  we  shall  have, 


(-a)'  =(+a«)X  (-«)  =  -<*•; 


(_a)>  =  (-f-a')X(-a)  - 
And  in  general, 


the  plus  sign  in  the  second  member  being  used  when  n  is  even, 
and  the  minus  sign  when  n  is  odd.  Hence, 

1.—  All  powers  of  a  positive  quantity  are  positive. 

2.  —  The  odd  powers  of  a  negative  quantity  arc  negative,  but  the 
even  powers  are  positive. 


POWERS    OF    MONOMIALS.  153 

211.  From  the  foregoing  principles  relating  to  the  involution 
of  a  monomial,  we  derive  the  following 

RULE.  I.  Raise  the  numeral   coefficients  to  the   required  power. 

II.  Multiply  the  exponent  of  each  letter  by  the  exponent  of  the 
required  power. 

III.  When  the  quantity  involved  is  negative,  give  the  odd  powers 

the,  minus  sign. 


EXAMPLES   TOR  PRACTICE. 

1.  Raise  x*  to  the  4th  power.  Ans.  cc". 

2.  Raise  y1  to  the  3d  power.  Ans.  y". 

3.  Raise  xn  to  the  6th  power.  Ans.  x*n. 

4.  Raise  xm  to  the  wth  power.  Ans.  a"1*1. 

5.  Raise  ax*  to  the  3d  power.  Ans.  a8x8. 

6.  Raise  ab*x*  to  the  2d  power.  Ans.  a*b*x*. 

7.  Raise  5«ax  to  the  3d  power.  Ans.  125aV. 

8.  Raise  8aa&3  to  the  2d  power.  Ans.  64a4&8. 

9.  Raise  — 4a  to  the  4th  power.  Ans.  256a4. 

10.  Raise  — 4a  to  the  3d  power.  Ans.  — 64a8. 

11.  Required  the  7th  power  of  — aax$.  Ans.  — al*x*1. 

12.  Required  the  4th  power  of  — 3ccP.  Ans. 
Find  the  values  of  the  following  indicated  powers : 

13.  (6a&3)8.  Ans. 

15.  (amb»y.  Ans.  a**bin. 

16.  (— a*")3.  Ans.  a3". 

18.  (— 3ac^)J.       t  Ans. 

20.  (— abc)m.  Ans.   ±ambmcm. 


154  INVOLUTION. 

212.  If  it  be  required  to  raise  am  to  the  mth  power,  we  shall 
have 

(am)m  =  amxw  =  am\ 

an  expression  which  denotes  that  power  of   a  whose  index  is  m*. 
If  we  put  m  =  3,  then  am  =  a". 

Expressions  like  the  above  may  frequently  occur  in  algebraic 
operations. 

EXAMPLES. 

Find  the  value  of  each  of  the  following  expressions  : 

1.  (x"y)*.  Ans.  xmnyn*. 

2.  (a-y)-  Ans.  xm*ymn. 

3.  (x^y.  Ans.  xm*. 

4.  \jx    j    .  Ans.  x 

5.  (-c"v~1y)TO+1.  Ans.  xm  ~lym+l. 

6.  (ab*cn  dn  )n.  Ans.  onin  cn  c?1  . 

POWERS  OF  FRACTIONS. 

213.  If  a  fraction  be  raised  to  any  power  by  multiplication,  both 
numerator  and  denominator  will  be  raised  to  the  same  power. 

a 
1 .  Required  the  3d  power  of  — 

a  \*       a       a       a 

Hence,  to  raise  a  fraction  to  any  power,  we  have  the  following 
RULE.  Raise  both  numerator  and  denominator   to  the  required 
power. 

EXAMPLES    FOR  PRACTICE. 

1.  Involve  •=-=  to  the  2d  power.  Ans.   ^-.* 

6tr  6V 

aa  "* 

2.  Involve   =~^  to  the  3d  power.  Ans. 


NEGATIVE    INDICES, 
3.  Involve =—  to  the  5th  power.  Ans.  — 


4.  Involve   --  to  the  4th  power. 

W 

5 

5.  Raise  —  •=-  to  the  6th  power. 


2xm 
G.  Raise  „         to  the  5th  power. 

7.  Raise  --  to  the  nth  power. 
ryz 


8.  Find    ( - 

9.  Find 


6)' 


155 

1024g"&B 

16807x> ' 

Ans.  —7—7- 


Ans. 

>. 
Ans. 


15625 


anl> 


.  ± 
Ant. 
Ans.  -TV 


DISCUSSION  OF  NEGATIVE  INDICES. 

314.  It  has  been  shown  in  previous  articles  that 

am 
awX«n=«m+n  '    —=«•""*>    and   (am)n     =  a""  , 

where  m  and  n  are  positive  whole  numbers.    It  remains  to  be  shown 

that  the  above  relations  hold  true  when  one  or  both  of  the  expo- 

inents  are  negative.     And  in  this  investigation  it  is  sufficient  to  re- 

\member  that  a  quantity  with  a  negative  exponent  is  equal  to  the  re- 

jciprocal  of  the  same  quantity  with  a  positive  exponent ;  (88,  2). 

I.  To  prove  that  amxan  =  am+n  universally,  m  and  n 
being  integers. 

1. — Suppose  one  of  the  exponents  to  be  negative ;  or  let 
n  =  — n'. 

2- — Suppose  both  exponents  are  negative ;  or  let 
m  =  — m'  and  n  =  — n'. 
1 


Then 


~   a1*'    X  a"'  ~ 


156  INVOLUTION. 

II.  To  prove  that  —  =  am-n  universally,  m  and  n  being 
integers. 
1. — Suppose  the  exponent  of  the  numerator  to  be  negative ;  or 

am       a-"' 


_ 

an          a"          «»»'+«  —  =          • 

2.  —  Suppose  the  exponent  of  the  denominator  to  be  negative  ; 
or  let  n  =  —  n'. 


Then       -  =         =  awX«B'  =  a"*4*'  =  a"-*. 

3.  —  Suppose  both  exponents  are  negative  ;  or  let 
m  —  —  m'   and   n  =  —  nr. 


III.  To  prove  that  (an)n  =  amn  universally,  m  and  n 
being  integers. 

1.  —  Suppose  n  to  be  negative  ;  or  let 


Then    (a-)»  =  («-)-' =  =  -7  =  a 

2. — Suppose  m  to  be  negative ;  or  let 


/  1  \A         1 

OT = c*-o-  =  5=)  =  5==  =  «-"= •»-. 


Then 


3. — Suppose  both  m  and  n  to  be  negative ;  or  let 
m  =  — mr  and  n  =  — ji'. 


Then  (a")"  =  (a-w/)-n/=  (~\      =  (^-)*  =  am/n'  =  a™ 

Hence,  in  all  algebraic  operations,  the  same  rules  will  apply  to 
negative  exponents  as  to  positive.  That  is,  if  two  powers  of  the 
same  quantity  be  given,  then  the  exponent  of  their  product  will  be 
equal  to  the  algebraic  sum  of  the  given  exponents,  and  the  exponent 
of  their  quotient  will  be  equal  to  the  algebraic  difference  of  the  giv- 

on  r*  YYWn  r»n  fa  * 


en  exponents 


POWERS    OF   POLYNOMIALS.  157 


EXAMPLES. 

5.  Find  the  value  of  each  of  the  following  expressions: 

1.  (a-7//)3.  Ans.  a~*l\ 

2.  (i~V)-a.  Ans.  Vc-4. 
8.    2xH"~i.                                                      Ans.     arm. 


4.  (4aw&-*)a. 

5.  (_catZ-*m4)\ 

C.  (Sa-'ay-1)-4.  -4ns. 

7.  C—o'V")"  -4**  ±a«~"'"- 

8.  (ar"1')1*"3.  4«s.  a;-*"1. 

9.  (4a'*-2)'  X  (a-V).  ^n«.  16a. 
10.  (aa"6--)aX(a-t"i"m)"*.  ^«*.  a-i"-4". 

POWERS  OF  POLYNOMIALS. 

S^IG.  A  polynomial  may  be  raised  to  any  power  by  actual  mul- 
tiplication Thus,  if  the  quantity  be  multiplied  by  itself,  the  prod- 
uct will  oe  the  second  power;  if  the  second  power  be  multiplied 
by  the  quantity,  the  product  will  be  the  third  power  ;  and  so  on. 
Hence  the  following 

RULE.  —  Multiply  the  quantity  by  itself  in  continued  multipli- 
cation, till  it  has  been  taken  as  many  times  as  a  factor  as  there  are 
units  in  the  exponent  of  the  required  power. 

NOTE.  It  may  be  well  to  observe  that  in  involution  we  may  often  reach 
the  same  result  by  different  processes.  Thus,  we  have  a6  =a5xa  = 


EXAMPLES    FOB    PRACTICE. 

Expand  the  following  expressions  : 


Ant. 

2.  (5z-y)».  Ans.  125*'— 

3.  (l-f-2x—  £$')'.  Ans.  l-J-4z—  2z»—  12z'-|-9z4. 

14 


158  INVOLUTION. 

4.  (3a+2b+cf. 

Am.  27a'+54a'6-f  27aac+36a&'+36a6c+8&8+9ac»-f  12ifc+ 
66c«-fc». 

5.  («+&)». 

^*s.  a'-f  7ae&+21a>&a+35a4&'-|-35a8&4-f  21a'i*4-  7a&'-f&T. 

6.  (x-y}\ 

Am.  x9  —  8zV-f-28*y  —  56xy  +  70xy—  56*y  +28u:y  — 


7.  (aV-'-fa-V)1.  4ns.  a4c~4-f  2-j-a~4c*. 

8.  (a'-f  1-fa-')8.     4ns.  a'-f  3a4+6aa+7H-6a-34-3a-44-a-$. 
9. 


POLYNOMIAL  SQUARES. 


.  We  have  seen  that  the  square  of  any  binomial  may  be 
written  without  the  labor  of  formal  multiplication,  (7O).  Thus, 
if  x  and  y  represent  the  terms  of  any  binomial,  then 


This  formula  for  a  binomial  square  furnishes  a  simple  rule  for 
writing  out  the  square  of  any  polynomial,  in  the  same  direct  manner. 
To  deduce  the  method,  let  it  be  required  to  square  the  polynomial, 


Put  x  =  a  and  y  =  l-{-c-\-d-\-e-{-  .....     Then  the  square  of  x-f  y 
will  be  equal  to  the  square  of  the  given  polynomial  ;  or 


And  the  three  parts  of  the  required  square  will  be 

of  =  a',  (1) 

2xy  =  2a6-f2ac+2a^-f2ae+  ____  ,  (2) 


Now  y  represents  a  polynomial ;  and  to  obtain  its  square,  we  must 
proceed  as  at  first.     Thus,  put  x'  =±  b  and  y'  —  c -\-d-\-e-\- 
Then  the  square  of  x'-fy'  will  be  equal  to  the  square  of  l-\- c- 
e-f- And  we  have 


POLYNOMIAL    SQUARES.  159 

x13  =  V,  (3) 

2x'y'  =  2bc+2bd+2be+  .  .  .  .  ,  (4) 

y"  =  (<:+<?+«+    ...)«. 

If  we  proceed  with  the  value  of  y'a  as  with  the  value  of  #a,  we 
shall  finally  obtain  all  the  parts  of  the  required  square. 

By  inspecting  equations  (1),  (2),  (3)  and  (4),  we  perceive  that  the 
required  square  will  assume  the  following  general  form  : 

(a+b+c+d+e+..  .  .)"  =  aa-f  2a(6-f-f-f  J+e+  .  .  .  .)  -f  i'-f- 
2&(c-HZ-j-e-f-  ____  )  -f-ca-j-2c  (d-\-e-\-  ____  ),  and  so  on.  Hence  to 
obtain  the  square  of  any  polynomial,  we  have  the  following 

Rule.  Write  the  square  of  each  term,  together  with  twice  the  pro- 
duct of  each  term  by  tJie  sum  of  all  the  terms  which  follow  itt  and 
reduce  the  result  if  necessary. 

EXAMPLES  FOE  PRACTICE. 

1.  Square  a-f-6-f  c.  Ans.  ai+2a&+2ac+&>+25c+<£ 

2.  Find  the  square  of  a-\-b+c-\-d. 

Ans.  aa-f2a6+2ac-f2ac?-f-ia-|-2 

3.  Find  the  square  of  a-\-l>-\-  c+d-\-e. 
Ans.  a2  +  2al  +  2ac  +2 


4.  Square  x  —  y+z.  Ans.  x*  —  2xy-}-  2xz-\-y*—  2yz-\-z*. 

5.  Find  the  square  of  a  —  26-{-3a5  —  c. 

Ans.  aa—  4a6-f6a96—  2ac-f46a—  12a&a+46c-f9aa6a—  6a5c+c*. 

6.  Find  the  square  of  1  —  a-j-a8  —  a1. 

Ans.  l_2a-|-3a'—  4a"+3a4—  2aB-}-ai. 

7.  Find  the  square  of  3ax-}-2a9  —  4xs—  5. 

Ans.  9aV+  12a'x—  24ax8—  30ax+4a4—  16asx«—  20a'-f  IQx* 
-f40xa+25. 

8.  Find  the  square  of  1  —  2x—  y*-\-xy  —  x*. 

Ans.  I—4x—2y*-\-  2xy-\-  2x*+4:xy*—4x*y  +  4x'+<y4  —  2xy'— 
2x3y  +  3;c2y2  +  x\ 

918.  In  a  future  section  we  shall  give  a  formula,  called  the 
Binomial  Formula,  by  means  of  which  any  power  of  a  binomial  may 
be  obtained  without  the  labor  of  multiplication. 


160  EVOLUTION 


EVOLUTION. 

219.  A  Root  of  any  quantity  is  one  of  the  equal  factors  -which, 
multiplied  together,  will  produce  the  given  quantity. 

220.  The  name  or  degree  of  a  root  corresponds  to  the  number 
of  equal  factors  into  which  the  quantity  is  supposed  to  be  divided. 
Thus, 

The  square  root  of  a  is  one  of  the  two  equal  factors  whose 
product  is  a. 

The  cube  root  of  a  is  one  of  the  three  equal  factors  whose 
product  is  a. 

The  fourth  root  of  a  is  one  of  the  four  equal  factors  whose 
product  is  a  ;  and  so  on. 

221.  Evolution  is  the  process  of  extracting  any  root  of  a  given 
quantity ;  it  is  the  converse  of  involution. 

222.  There  are  two  methods  of  indicating  evolution  : 
1st.  By  the  radical  sign,  j/. 

When  this  method  is  employed,  the  name  or  degree  of  the  root 
is  denoted  by  a  figure  or  letter  written  above  the  radical,  called  the 
index  of  the  root.  Thus  \fa  denotes  the  cube  root  of  a;  and  \/a 
denotes  the  fourth  root  of  a.  When  no  index  is  written,  2  is  un- 
derstood. Thus  i/x  denotes  the  square  root  of  x,  and  signifies 
the  same  as  \fx. 

2d.  By  fractional  exponents. 

To  explain  the  origin  of  this  method  of  indicating  roots,  we 
observe  that  a  quantity  is  raised  to  any  power,  by  multiplying  its 
exponent  by  the  exponent  of  the  required  power.  Conversely,  any 
root  of  a  quantity  may  be  obtained,  by  dividing  the  exponent  of  the 
quantity  by  the  index  of  the  required  root.  Thus,  the  cube  root  of 
a,  or  a1,  is  written  a  ,  and  the  cube  root  of  aa  will  be  a  . 

Hence,  a  fractional  exponent  may  be  analyzed  as  follows  : 

1. —  The  numerator  denotes  the  power  of  the  quantity ,  whose  root 
is  to  be  extracted. 

2. —  The  denominator  sJiows  what  root  of  that  power  is  to  be 
extracted: 


BOOTS    OP    MONOMIALS.  101 

333.  The  two  methods  of  indicating  roots  may  be  illustrated 
by  equivalent  expressions,  as  follows  : 

I/a,  or  a  ,  denotes  the  square  root  of  a ; 
Va,  or  a*,       "        "      cubo      "  < "  a; 

Va,  or  a",      "         "       nth       "     "  a. 
And  if  am  represent  any  power  of  a,  then 

m 

I/a™,  or  a~,  denotes  the  square  root  of  am ; 
Vam,  ora^,       "         "     cube      «     "  a"; 

m 

VaM,ora%       «         «      wth       "     "  aw. 

334.  A  Surd  is  a  root  which  cannot  be  exactly  obtained ;  as 

,  %  Va9  or  I/a3— 2a6. 

A  surd  is  called  an  irrational  quantity,  while  a  root  which  can 
be  exactly  obtained  is  called  a  rational  quantity.  A  root  will  bo 
rational  when  the  given  quantity  is  a  perfect  power  corresponding 
in  degree  to  the  required  root ;  otherwise  it  will  be  a  surd. 

The  root  of  a  number  which  is  an  imperfect  power,  may  always 
be  obtained  approximately.     Thus,  j/6  is  a  surd ;  but  we  have 
^6=2.44,  nearly;  for  (2.44)9=5.9336. 

33t>.  An  Imaginary  root  is  one  which  is  known  to  be  impossi- 
ble on  account  of  the  sign  of  the  given  quantity.  Thus,  the  square 
root  of  — aa,  or  V — aa,  is  impossible,  since  no  quantity  raised  tn 
the  second  power  will  produce  — a9.  A  root  which  is  not  imaginary 
is  said  to  be  real. 


ROOTS    OP    MONOMIALS. 


33O.  It  has  already  been  shown  thnt  the  root  of  a  simple 
algebraic  quantity  may  be  expressed  by  dividing  the  exponent  of 
the  quantity  by  the  index  of  the  required  root  (333).  And  it 
is  evident  that  if  the  exponent  of  the  quantity  will  not  exactly  con- 
tain the  index  of  the  required  root,  the  result  must  be  a  surd. 
14*  L 


162  EVOLUTION. 


.  We  have  seen  that  a  quantity  composed  of  several  fac- 
tors, is  raised  to  any  power  by  involving  each  factor  separately  to  the 
required  power  ;  (2O8).  Conversely,  we  shall  obtain  the  root  of 
a  quantity  by  extracting  the  root  of  each  factor  separately.  Thus, 
if  abc  ..../»;  represent  the  product  of  any  number  of  factors,  then 


____  k  =  Va  V&  Vc  ____  V&  ; 
or,  with  fractional  exponents, 

-L         .1  JL  JL  JL 

(abc.  .  .  .&)»  =  anbnc»  .  .  .  .&*. 
That  is, 

The  nth  root  of  the  product  of  two  or  more  factors  is  equal  to 
the  product  of  the  nth  roots  of  the  factors. 

228.  There  are  certain  properties  of  roots  which  depend  upon 
the  law  of  signs  in  involution  : 

1.  —  Every  odd  root  of  a  quantity  is  real,  and  has  the  same  sign 
as  the  quantity  itself. 

For,  any  positive  quantity  raised  to  an  odd  power  is  positive  ;  and 
any  negative  quantity  raised  to  an  odd  power  is  negative;  (21O). 

2.  —  Every  even  root  of  a  positive  quantity  is  real,  and  may  be 
either  positive  or  negative. 

For,  either  a  positive  or  negative  quantity  raised  to  an  even  pow- 
er is  positive  ;    (21O). 

3.  —  Every  even  root  of  a  negative  quantity  is  imaginary. 

For,  no  quantity,  whether  positive  or  negative,  raised  to  an  even 
power,  will  give  a  negative  result. 

££59.  From  the  principles  now  established,  we  have  the  follow- 
ing rule  for  extracting  the  roots  of  monomials  : 

RULE.  I.  Extract  the  required   root  of  the  numeral  coefficients 
for  a  new  coefficient. 

II.  Divide  the  exponent  of  each  literal  factor  by  the  index  of  the 
required  root. 

III.  Prefix  the  double  sign,  ±,  to  all  even  roots}  and  the  minus 
sign  to  the  odd  roots  of  a  negative  quantity. 

NOTES.  1.  When  the  required  root  of  any  factor  is  a  surd,  it  may  be 
indicated  either  by  a  fractional  exponent,  or  by  the  radical  sign. 

2.  The  root  of  a  fraction  may  be  obtained  by  taking  the  root  of  the  nu 
merator  and  denominator  separately. 


ROOTS    OP    MONOMIALS.  163 


EXAMPLES  FOR  PRACTICE. 

1.  What  is  the  square  root  of  49aa:e4  ?  .4ns.   ±7a;c'. 

2.  What  is  the  square  root  of  25c106a  ?  Ans.   ±5c56. 

3.  What  is  the  square  root  of  144aVa:y?  Ans.  ±12acVy. 

4.  What  is  the  cube  root  of  125a9  ?  -A»*.  5a. 

5.  What  is  the  cube  root  of  —  64x"  ?  Ans.  —  4a;a. 

6.  What  is  the  cube  root  of  —  216ay  ?  Ans.  —  Say*. 

7.  What  is  the  cube  root  of  729aV2  ?  Ans.  9aV. 

8.  What  is  the  4th  root  of  256aV  ?  Ans.   ±4ax*. 

9.  Find  the  4th  root  of  16a.                  Ans.  ±2a*,  or  ±2Va. 

10.  Find  the  cube  root  of  27a*^.               Ans.  Sa3./*,  or3Va*x. 

11.  Find  the*5th  root  of  —  82x'y-    J.n*.  —2x'i/*,or—2x*  Vy4. 

12.  Find  the  nth  root  of  a*nlm.  Ans.  a'b  n  . 

13.  Find  the  square  root  of  81a-46*.  Ans.   ±9a~ai3. 

X^.  j.  iuvx  vnv;  vt*^  iv-^  v*    ----  ^-        ~      .  -----           —      ~       . 

15.  Find  the  5th  root  of  243cr6&-10.  Ans.  3a~J&-a. 

16.  Find  the  mth  root  of  cry*-  Ans.  anym. 

17.  Find  the  nth  root  of  afV»V4.  ^Lws.  ^"92ni. 

^  4a'x4  2x« 

18.  Required  the  square  root  of  -^-5-  -  -4n«.    ±  -g- 


fl 

19.  Required  the  cube  root  of    g  ,  $••  Ans.  -  —  4« 

200aT  5a8 

20.  Required  the  square  root  o*~9' 


21.  Required  the  nth  root  of  f^!T.  Ans.    —. 

a  a 


_ 

22.  Required  the  nth  root  of-r—  •  ,4ns.    anb  nc  n. 

23.  Find  the  square  root  of  (a  —  x)*/*  ^4ns.   ±  (a  —  x}y*. 

24.  Find  the  cube  root  of  (x—  I)3  (x+l)§.  -4«s.   (**—  1)  (^+1 

25.  Find  the  square  root  of  x*y*  (x  —  y)*.      Ans.  ±(x*y9—  xyz\ 


164  EVOLUTION. 


SQUARE  ROOT  OF  POLYNOMIALS. 

23O.  To  deduce  a  rule  for  the  extraction  of  the  square  root  of 
a  polynomial,  let  us  first  observe  how  the  square  of  any  binomial,  aa 
a-f-&,  is  formed.  We  have 


And  the  last  two  terms  may  be  written  as  follows  : 


Let  us  now  consider  how  the  process  of   involution  may  be  re- 
versed, and  the  root,  a-j-&,  derived  from  the  square. 

Extracting  the  square  root  of  a',  OPERATION. 
we  obtain  a,  the  first  term  of  the  root. 

Taking  aa  from  the  whole  expression,  a*-[-2ab-\-l*\a-{-b 

we  have  2aZ»-j-&2,  or  (2a-f  &)&,  for  a  a9 


remainder.      Dividing  the  first  term     2a-f-6  2ab-\-  u* 

of  this  remainder  by  2a,  as  a  partial  2aZ»-f-&a 

divisor,  we  obtain  6,  which  we  place 

in  the  root,  and  also  at  the  right  of  the  2a  to  complete  the  divisor, 

2a-|-Z>.     Multiplying  the  complete  divisor  by  &,  and  subtracting  the 

product  from  the  dividend,  we  have  no  remainder,  and  the  work  is 

finished. 

By  the  same  process  continued,  we  may  extract  the  root  of  any 
quantity  that  is  a  perfect  square.  To  establish  the  rule  in  a  general 
manner,  let 


represent  any  polynomial.  By  a  previous  article,  the  square  of 
this  polynomial  consists  of  the  square  of  each  term,  together  with 
twice  the  product  of  each  term  Li/  all  the  terms  which  follow  it  ; 
and  the  square  may  be  written  as  follows  : 


And  it  is  evident  that  if  the  root,  a-\-l-}-c-\-d.  .  .  .,  is  arranged 
according  to  the  powers  of  some  letter,  the  square  will  also  be 
arranged  according  to  powers  of  the  same  letter. 

We  may  now  derive  the  root  from  the  square,  in  the  following 
manner  : 


SQUARE    ROOT   OF   POLYNOMIALS.  165 

OPERATION. 

|rt-j_54-e-fd5 ,  root 

a* +2ab+2ac+2ad +b*+2bc+2bd • 

a*     ' 

2ab+2ac+2ad +b*+2bc+2bd. .  & 

2ab  +  Z»* 

+2ac + 2ad +2bc+2bd 

2ac 


,-2b+2c+d  2ad +2bd. . . .       +2cd +d* 

2ad  +2bd  +2cd        +d? 


We  find  a  as  in  the  former  example,  and  take  its  square  from  the 
whole  expression.  We  then  divide  the  first  term  of  the  remainder 
by  2a,  and  write  the  quotient,  b,  in  the  root,  and  also  in  the  divisor. 
We  then  multiply  the  complete  divisor  by  b,  subtract  the  product 
from  the  first  remainder,  and  thus  obtain  a  new  dividend.  Then 
writing  2a-f-26  for  a  partial  divisor,  we  find  c  in  the  same  manner 
as  we  found  b ',  and  thus  we  continue  till  the  work  is  finished. 

If  we  examine  the  several  subtrahends,  taking  the  terms  diag- 
onally in  the  operation,  we  shall  find  a2,  2ab,  2ac,  2«c7,  etc. ;  &*, 
2&c,  2bd,  etc. ;  c',  led,  etc. ;  d1 ,  etc.  That  is,  we  have,  in  the  ope- 
ration, the  square  of  each  term  of  the  root,  together  with  twice  the 
product  of  each  term  by  all  the  terms  ichlch  follow  it.  Thus  we 
have  exactly  reversed  the  process  of  forming  a  polynomial  square. 
Hence  the  following  general 

RULE.  I.  Arrange  the  terms  according  to  the  powers  of  some 
letter,  and  write  the  square  root  of  the  first  term  in  the  quotient. 

II.  Subtract  the  square   of   the  root   thus  found  from  the  given 
quantity,  and  bring  down  two  or  more  terms  for  a  dividend. 

III.  Divide  the  first  term  of  the  dividend  by  twice  the  root  already 
found,  and  write  the  result  both  in  the  root  and  in  the  divisor. 

IV.  Multiply  the  divisor,  thus  completed,  by  the  term  of  the  root 
last  found,  subtract  the  product  from  the  dividend,  and  proceed  with 
the  remainder,  if  any,  as  before. 

NOTE — According  to  the  law  of  signs  in  evolution,  every  square  root 
obtained  will  still  be  a  root,  if  the  signs  of  all  its  terms  be  changed. 


•166  EVOLUTION. 


EXAMPLES  FOR  PRACTICE. 

1.  What  is  the  square  root  of  aa-|-2a&-|-2ac-f-69-f-26c-|-c9  ? 

Ans.  a-f-6-j-c. 

2.  What  is  the  square  root  of  a*  —  6a26+4a3  +  96" — 126  +  4  \ 

Am.  a2— 36+2. 

3.  What  is  the  square  root  of  x*+ 4x*-f2z4— 2x'-f-5x9 — 2x 


4.  What  is  the  square  root  of  1—  2o-j-3aa—  4a'-f  3a4—  2a'-f- 

Ans.  1—  a-f-aa—  a\ 

5.  What  is  the  square  root  of  4a*l9—  1  2a'&9-f8a*&'-|-9a9&a— 


6.  What  is  the   square  root   of    9x"—  30x6 


7.  What  is  the  square  root  of  a4—  6a*6c-|-4aW  —  2a2 

Ans.  aa 


8.  What  is  the  square  root  of  a4—  a'6-|-  -^-  --  —  -  -f  —  ? 


Ans,  a»_._  +  _ 


9.  What  is  the  square  root  of  x*— 

Ans. 

10.  What  is  the  square  root  of  a*l~*—  10ai-1+27  — 

Ans.  al~l—  S+a^ft. 

11.  What  is  the  square  root  of  a4m+6af~c"4-llaamc8n+6a''lc8w-f- 
?  Ans.  a'm-f3amcn-|-can. 


SQUARE  ROOT  OF  NUMBERS. 

.  In  order  to  discover  the  process  of  extracting  the  squaro 
root  of  a  number,  it  is  necessary  to  determine 

1st.  The  relative  number  of  places  in  a  number  and  its  square 
root. 

2d.  The  local  relations  of  the  several  figures  of  the  root  to   the 
periods  of  the  number. 


SQUARE    ROOT    OF    NUMBERS.  167 

3d.  The  law  by  which  the  parts  of  a  number  are  combined  in  the 
formation  of  its  square. 

222.  The  relative  number  of  places  in  a  given  number  and  its 
square  root  may  be  shown  by  illustrations,  as  follows  : 

I3  =  1  I1  =  1 

9'=  81  10'=  1,00 

99'=  98,01  100'  =  1,00,00 

999'  =         99,80,01  1000'  =      1,00,00,00 

From  these  examples  we  perceive  that  a  root  consisting  of  1  place 
may  have  1  or  2  places  in  the  square  j  and  that  in  all  cases  the  addi- 
tion of  1  place  to  the  root  adds  2  places  to  the  square.  Hence, 

If  we  point  off  a  number  into  tico-fiyure  periods,  commencing  at 
the  right  hand,  the  number  of  periods  will  indicate  the  number  of 
places  in  the  square  root. 

233.  If  any  number,  as  2345,  be  decomposed  at  pleasure,  the 
squares  of  the  parts,  beginning  with  the  highest  order,  will  be  rela- 
ted in  local  value  as  follows : 

2000' =     4  00  00  00 

23002  =     5  29  00  00 

2340' =     5  47  56  00 

2345'  =     5  49  90  25.  Hence 

The  square  of  the  first  fyure  of  the  root  is  contained  wholly  in 
the  first  period  of  the  power  ;  the  square  of  the  first  tico Jiynres  of 
the  root  is  contained  wholly  in  the  first  two  periods  of  the  power  ; 
and  so  on. 

234.  If  the  figures  of  a  number  bo  separated  into  two  parts, 
and  written  with  their  local  value,  we  may  then  form  the  square  of 
the  number  by  the  formula  for  a  binomial  square.     Thus, 

76  =70  -|-  6.     And  if  we  put  a  =  70  and  6  =  6,  then 
a-\-b  =  76 ;  and  we  shall  have 

a'  =  4900  ' 
2ab  =   840 
V  =      36 


a*+2ab-\-b*=  5776=76' 

Hence,  the  binomial  square  may  be  used  as  a  formula  for  extracting 
the  square  root  of  a  number. 


168  EVOLUTION. 

1.  Let  it,  be  required  to  extract  the  square  root  of  5776. 
There  are  two  periods  in  the  num-  OPERATION. 

her,  indicating  that  there  will  be  two 

places  in  the  root.     As  the  square  of  5  /   <  6  [76 

the  tens  is  contained  wholly  in  the  aa,  40  00 

first  period,  (233),  we  first  seek  the  2«,     140      8  76 

greatest  perfect  square  in  57.  This  2a-\-b,  146  8  76 
we  find  to  be  49,  the  root  of  which  is  7,  the  first  figure  of  the  root 
sought.  Hence  we  have  a  =  70,  and  subtracting  a2,  or  4900,  from 
the  entire  number,  we  have  876  for  a  remainder,  which  must  be 
equal  to  (Za-\-l)b ;  (23O).  Dividing  the  remainder  by  the  partial 
divisor,  2«,  or  140,  we  have  b  =  6,  the  second  figure  of  the  root. 
Completing  the  divisor,  we  have  2a-\-b  =  146 ;  whence  (2a-f-^)X 
b  =  876,  and  the  work  is  complete. 

It  is  obvious  that  we  may  omit  ciphers,  and  still  employ  the 
figures  with  their  proper  local  values,  in  the  operation.  It  will  not 
then  be  necessary  to  form  the  partial  divisor  separate  from  the  com- 
plete divisor. 

If  the  given  number  consists  of  more  than  two  periods,  we  may 
extract  the  two  superior  figures  of  the  root  from  the  first  two 
periods,  (233),  bringing  down  another  period  to  the  remainder. 
Then  a  in  the  binomial  formula  will  represent  the  part  of  the  root 
already  found,  considered  as  tens  of  the  next  inferior  order  ;  and 
so  on. 

2.  Required  the  square  root  of  22657C. 

OPERATION. 

ft 

22  6*5  76  |  476,  Ant. 
16 

87      665 
609 


946      5676 
5676 


Having  found  47,  the  square  root  of  the  first  two  periods,  we 
bring  down  the  last  period,  and  have  5676  for  a  new  dividend.    "We 


SQUARE    ROOT    CI-    NUMBERS.  169 

then  take  :."a  =  47x2  —  94  for  a  partial  divisor,  whence  we  obtain 
I  —  6,  the  last  figure  of  the  root.  We  should  observe  that  by 
pimply  doubling  the  7  in  the  87.  we  may  obtain  94,  the  new  trial 
divisor. 

From  these  principles  and  illustrations,  we  have' the  following 
RULE.  I.  Point  off  the  given  number  into  periods  of  two  figures 
each,  counting  from  unit1  s  place  toward  the  left  and  right. 

II.  Find  the  greatest  square  number  in  the  left-hand  period,  and 
write  its  root  for  the  first  figure  in  the  root  sought ;    subtract  the 
square  number  from  the  left-hafid  period,  and  to  the  remainder  bring 
down  the  next  period  for  a  dividend. 

III.  Al  the  left  of  the  dividend  write  twice  the  first  figure  of  the 
root,  for  a  trill  divisor  •  divide   the  dividend,  exclusive  of  its  right- 
hand  figure,  by  the  trial  divisor,  and  write   the  quotient  for  a  trial 
figure  in  the  root. 

IV.  Annex  the  trial  figure  of   the  root  to  the  trial  divisor  for  a 
complete  divisor  ;  multiply  the  complete  divisor  by  the  trial  figure  in 
the  root,  subtract  the  product  from  the  dividend,  and  to  the  remainder 
bring  down  the  next  period  for  a  new  dividend. 

V.  Take  the  last  comjrtcte  divisor,  doubling  its  right-hand  figure, 
for  a  new  trial  divisor,  with  which  proceed  as   before,   tilt  the  work 
is  finished. 

NOTES.  1.  If  there  is  a  remainder  after  all  the  periods  have  been 
brought  down,  annex  periods  of  ciphers,  and  continue  the  operation  to  as 
many  decimal  places  as  are  required. 

2.  If  the  denominator  of  a  fraction  is  not  a  perfect  square,  the  fraction 
may  be  first  reduced  to  a  decimal,  and  its  root  then  taken. 


EXAMPLES    FOR   PRACTICE. 

1.  What  is  the  square  root  of  7225  ?  Ans.  85. 

2.  What  is  the  square  root  of  108241  ?  Ans.  329. 

3.  What  is  the  square  root  of  651249  ?  Ans.  807. 

4.  What  is  the  square  root  of  9741  CO  ?  Ana.  987. 
:>.  What  is  the  square  root  of  50985G1?  Ans.  2258. 

15 


170  EVOLUTION. 

6.  What  is  the  square  root  of  6634.1025  ?  Ann.  81.45. 

7.  What  is  the  square  root  of  1812886084?  Ans.  42578. 

8.  What  is  the  square  root  of  .339889  ?  Ans.  .583. 

9.  What  is  the  square  root  of  .00524176?  Ans.  .0724. 

10.  What  is  the  square  root  of  477  ?  Ans.  21.8403-f . 

11.  What  is  the  square  root  of  11.09  ?  Ans.  3.3301G+. 

12.  Required  the  square  root  of  71,3gsV  ^'ts-  73oV 

13.  Required  the  square  root  of  73^50^7.  Ans.  TTjTn7. 

14.  Required  the  square  root  of  %$$•  Ans.  j7T. 

15.  Required  the  square  root  of  5$.  Ans.  2.3604+. 

CONTRACTED    METHOD. 

2«S«I.  When  the  required  root  is  a  surd,  the  work  may  be 
abridged  by  the  method  of  contracted  decimal  multiplication.  To 
insure  a  correct  result,  each  contracted  divisor  should  contain  at 
least  one  redundant  pl'.u -e — that  is,  one  place  more  than  is  necessary 
to  produce  the  required  order  of  units  in  the  product.  This  figure 
should  be  multiplied  mentally,  and  the  tens  (increased  by  1  when 
the  units  are  5  or  more)  carried  to  the  product  of  the  next  figure. 

To  illustrate  this  principle,  let  it  be  required  to  divide  28337  by 
53194,  correct  to  3  decimal  places. 

In    multiplying   the    first   divisor,   of      53194)  28337  (.5327 
which  the  last  figure,  4,  is  treated  as  re-  26597 

dundant,  we  say  5  times  4  are  20,  and       5319        1740 
reserve  the  2  tens  for  the  next  partial  1595 

product;  then,  5  times  9  are  45,  and  2       532  145 

tens  added  make  47,  and  we  write  the  106 

unit  figure  of  this  result  for  the  first  in       53  39 

the  contracted  product.     In  multiplying  37 

the  second  divisor,  5319,  we  have  9x3 

=  27 ;  hence  there  will  be  3  tens  to  carry,  because  27  is  nearer  30 
than  20.  The  third  divisor  is  532,  one  unit  being  carried  to  531 
of  the  preceding  divisor,  because  the  rejected  figure,  9,  is  greater 
than  5. 


SQUARE    ROOT    OF    NUMBERS. 


171 


1.  Required  the  square  root  of  7.12  correct  to  six  decimal  places. 

We  continue  the  operation  as  usual 
until  we  have  obtained  the  dividend, 
1776.  At  this  point  we  omit  the  pe- 
riod of  ciphers,  and  consider  533  as 
the  divisor ;  and  in  multiplying  by  3, 
the  new  root  figure,  we  carry  the  1  ten  46 

from  the  product  of  the  redundant 
figure  6,  and  1  also  from  the  8  units  526     3600 

in  this  product,  making  1601  for  the  3156 

first  contracted  product.     After  this  5328     44400 

we  drop  one  figure  from  the  right,  to  42624 

form  each  successive  divisor,  and  thus  5336       1776* 

continue  till  the  work  is  finished.  1601 


OPERATION. 

12.668333  ±  Am. 
?;120000 
4 

312 
276 


534 


160 


53 


15 
16 


It  will  be  observed  that  there  are  as  many  figures  in  the  root  thus 
obtained,  as  there  are  in  the  assumed  number. 

From  this  illustration,  we  have  the  following 

RULE.  I.  If  necessary,  annex  periods  of  ciphers  to  the  (/wen 
number,  and  assume  as  many  figures  as  there  are  places  required 
in  the  root  ;  then  proceed  in  the  usual  manner  until  all  the  assumed 
figures  have  been  brought  down. 

II.  form  the  next  trial  divisor  as  usual,  but  omit  to  annex  to  it 
the  trial  figure  of  the  root,  reject  one  figure  from  the  right  to  form 
each  subsequent  divisor,  and  in  multiply  ing  regard  the  right-hand 
figure  of  each  contracted  divisor  as  redundant. 

NOTE. — If  a  rejected  figure  is  5  or  more,  increase  the  next  figure  at 
the  left  by  1. 

EXAMPLES. 

1.  Find  the  square  root  of  56  correct  to  7  decimal  places. 

Ans.  7.4833147-f . 

2.  Find  the  square  root  of  14  correct  to  7  decimal  places. 

Ans.  3. 7416573 +  . 


172  EVOLUTION. 

3.  Find  the  square  root  of  18  correct  to  4  decimal  places. 

Ans.  4.2426+. 

4.  Find  the  square  root  of  19  correct  to  6  decimal  places. 

Ans.  4.358898+. 

5.  Find  the  square  root  of  52.463  correct  to  7  decimal  places. 

Ana.  7.2431346+. 

6.  Find  the  square  root  of  7  correct  to  8  decimal  places. 

Ans.  2.64575131  -f. 

7.  Find  the  value  of  53  correct  to  5  decimal  places. 

Ans.  11.18034+. 


CUBE    ROOT    OF    POLYNOMIALS. 

236.  We  may  deduce  a  rule  for  extracting  the  cube  root  ol  a 
polynomial  in  a  manner  similar  to  that  pursued  in  square  root,  Ly 
analyzing  the  combination  of  terms  in  the  binomial  cube. 

If  the  binomial,  a+6,  be  cubed,  we  have  „ 


We  will  now  consider  how  the  process  may  be  reversed,  and  the 
root  extracted  from  the  power.  We  observe 

1st.  That  the  first  term  of  the  root  may  be  obtained  by  taking 
the  cube  root  of  the  first  term  of  the  power.  Thus, 

Va*  =  a. 

2d.  The  second  term  of  the  root  may  be  found  by  dividing  the 
second  term  of  the  power  by  three  times  the  square  of  the  first 
term  of  the  root.  Thus, 

=  b. 


3d.  The  last   three  terms   of  the  power  may  be  factored,  and 
written  as  follows  : 


or 

Thus  we  see  that  if  to  the  trial  divisor,  3  a2,  we  add  a  correction, 
3a&+&'2.  or  (3a+6)6,  the  result  will  be  a  complete  divisor,  which 
multiplied  by  A,  will  give  the  last  three  terms  of  the  power. 

Hence,  the  whole  operation  of  extracting  the  root,  a+6,  from 
the  cube,  as+3a26+3«&2+63,  may  be  written  as  follows  : 


CUBE   BOOT   OF   POLYNOMIALS.  173 


OPERATION. 


3a8 


Having  found  a,  the  first  term  of  the  root,  we  take  its  cube  from 
the  whole  expression,  and  obtain  3a*6-|-3a&a-|-&8.  Dividing  the 
first  term  of  this  remainder  by  3 a2,  we  obtain  6,  the  second  term  of 
the  root.  To  complete  the  divisor,  we  first  write  the  quantity  3a-|-&; 
and  multiplying  this  by  6,  we  have  3aZ>-j-&9,  which  added  to  the 
trial  divisor,  gives  3a2-f-3a&-{-62,  the  complete  divisor.  Multiplying 
this  by  />,  and  subtracting  the  product  from  the  dividend,  there  is 
no  remainder,  and  the  work  is  complete. 

297.  To  recapitulate, -we  may  designate  the  quantities  employed 
in  the  foregoing  operation,  as  follows : 

Trial  divisor,  3a*  "\ 

First  factor  of  correction,  3a-\-b  ( 

^  ( G) 
Correction  of  trial  divisor,  3ab-\-b*  C 

Complete  divisor,  3a»_j_3a&_j_ia  J 

338.  Next,  suppose  there  are  three  terms  in  the  root,  as  a-j-6 

+<-'• 

Assume  s  =.  a-\-b;  then  s-\-c  =  a-j-6-j-c;  and  we  have 

(s+c)3  =  s3+3sac+3scs+c8. 

If  we  proceed  as  in  the  last  example,  we  shall  obtain  a-\-b,  or 
that  part  of  the  root  represented  by  s,  and  subtract  its  cube  from 
the  whole  expression.  There  will  then  be  left  3sac-j-3$ca-|-c3,  which 
may  be  factored  and  written 

(3.sa-f3sc-fca)r*    or    {3sa-j-(3s-f-c)c}c. 

And  we  perceive  that  3x2  will  be  the  new  trial  divisor  to  obtain 
c,  and  that  (3s-f-c)c  will  be  the  new  correction. 

The  value  of  3sa,  or  3(a-|-&)a,  may  be  obtained  by  multiplication. 
It  will  be  more  convenient,  however,  to  derive  it  by  the  addition  of 
three  quantities  already  used  in  the  operation.  Thus, 

15* 


174  EVOLUTION. 

Last  complete  divisor,  3a*-f-3a&-{-&a          -\ 

Last  correction,  Sab-\-l*          \.       n\ 

Square  of  last  term  of  the  root,  6*          J 

3.s3  =  3(a-j-£)a  =  3a3-f  6ai-|-36a 
Let  it  now  be  required  to  find  the  cube  root  of  the  polynomial 


OPEBATION. 

la^-faj—2,  root. 


x* 


3x*+x 


3z6-3.z4-llz8+62:2+12.i:-8 
x3 


Having  arranged  the  polynomial  according  to  the  exponents  of  x, 
we  proceed  as  in  the  former  example,  and  obtain  x2,  the  first  term 
of  the  root,  3x*— 3x4— Ilx3-j-6x8-f-  12x— 8  the  first  remainder, 
3x4  the  trial  divisor,  and  x  the  second  term  of  the  root.  To  com- 
plete the  trial  divisor  according  to  formula  (a),  we  write  three  times 
the  first  term  of  the  root  plus  the  second,  or  3x3-|-a:,  for  the  first 
factor  of  the  correction.  Whence  we  have  (3x8-|-x)x,  or  Sx'-j-cc2, 
for  the  correction;  3x4-f-3x*-j-x*  for  the  complete  divisor;  (3x4-j- 
Sos'-f  a;1)*,  or  3xB-f-3z4-fV,  for  the  product;  and—  6x4— 12xs-f- 
Gxa-|-12x — 8  for  the  new  dividend. 

To  form  the  new  trial  divisor  according  to  formula  (&),  we  have 
(3x4-|-3x8-fx9)  -f-  (3x'-|-a;3)  -j-*2  =  3x4-f 6xs-f 3.x2;  whence,  by 
division,  we  obtain  — 2  for  the  third  term  of  the  root.  To  com- 
plete the  new  trial  divisor,  we  have  for  the  first  factor  of  the  cor- 
rection, 3(xa-|-.T) — 2  =  3xa-f-8^ — 2.  This  may  be  obtained  in  the 
operation  from  the  former  factor  3#a-j-:r,  by  simply  multiplying  its 
second  term  by  3,  and  annexing  the  — 2.  We  now  find  the  correc- 
tion, complete  divisor,  and  product  as  before,  and  the  work  is 
finished.  It  is  evident  that  three  or  more  terms  of  the  root  will 
sustain  the  same  relation  to  the  next  succeeding  term,  that  the  first 
sustains  to  the  second,  or  the  first  and  second  to  the  third. 


CUBE    ROOT    OF    POLYNOMIALS.  175 

23O.  From  the  foregoing  analysis  we  derive  the  following 
RULE.  I.  Arrange  the  polynomial  according  to  the,  powers  of  some 
letter,  and  write  the  cube  root  of  the  first  term  in  the  quotient ;  sub- 
tract the  cube  of  the  root  thus  found  from  the  polynomial,  and  ar- 
range the  remainder  for  a  dividend. 

II.  At  the  left  of  the  dividend  write  three  times  the  square  of  the 
root  already  found,  for  a  trial  divisor  ;  divide  the  first  term  of  the 
dividend  by  this  divisor,  and  write  the  quotient  for  the  next  term  of 
the  root. 

III.  To  three  times  the  first  term  of  the  root  annex  the  last  term, 
and  icrite  the  result  at  the  left,  and  one  line  below,  the  trial  divisor  ; 
multiply  this  result  by  the  last  term  of  the  roof,  for  a  correction  of 
the  trial  divisor  ;  add  the  correction,  and  the  result  will  be  the  com- 
plete divisor. 

IV.  Multiply  the  complete  divisor  by  the  last  term  of  the  root, 
subtract  the  jtrotluct  from  the  dividend,  and  arrange  the  remainder 
for  a  new  dicidend. 

V.  Add  together  the  last  complete  divisor,  the  last  correction,  and 
the  square  of  the  last  term  of  the  root,  for  a  new  trial  divisor  ;  and 
by  division  obtain  another  term  of  the  root. 

VI.  Take  the  first  factor  of  the  last  correction  with  its  last  term 
multiplied  by  3,  and  annex  to  it  the  last  term  of  the  root,  for   the 
first  factor  of  the  new  correction  ;  with  which  proceed  as  before, 
till  the  work  i$  finished. 

EXAMPLES     FOR    PRACTICE. 

1.  What  is  the  cube  root  of  27«8-f  108as-{-144a-f  64  ? 

Ans.  3a-}-4. 

2.  What  is  the  cube  root  of  a  6-j-6x6— 40xs4-96x— 64  ? 

Ans.  x*-\-1x— 4. 

3.  What  is  the  cube  root  of  Sxe— 36x6-f66.r4— 63x8-f33x2— 9x 
-f-1?  Ans.  2xa— Sx+l. 

4.  What  is  the  cube  root  of   afl-f  9a8&+24aV/l-|-9a>&'— 24a2Z>4 
-f9^s— Z/'?  Ans.  a*+3ab—b*. 

5.  What  is  the  cube   root  of   a9 — 6a8-|-27«7— 74afl+159a6— 
234a4-j-257a8— 174a8-j-60«— 8  ?  Ans.  a3— 2aa+5a— 2. 


176  EVOLUTION. 

-  G.  What  is  the  cube  root  of  *'— 3x8+6.tT— 10;c'-f  12x§— 12x« 
-flOx3— G.r'+Sx— 1?  ^l«s.  a;1— *'-f.r— 1. 

7.  What  is  the  cube  root  of  8«a  —  12«si  -f  SGa'ic  -f  6a*&s  — 
SGaVc— aV4-54a&V+9a*63c— 27a6V+27£V? 

-Aws.  2a— rti-f SJc. 

195x4  195  r1 

8.  What  is  the  cube  root  of  *'— 12**+  — j 70;c'-J-  -^j- 

3*       1  ?  1 

-T^64?  •4z+r 

9.  What  is  the  cube  root  of  o;8-}-Gx8—  64o:e— 962r6-j-192jc*4- 


Root*. 

Cubes. 

Roots. 

1 

r 

1 

9 

729 

10 

99 

970,299 

100 

999 

997,002,999 

1000 

CUBE  ROOT  OF  NUMBERS, 

24O.  To  establish  a  rule  for  extracting  the  cube  root  of  a  num- 
ber, we  must  first  ascertain  the  relative  number  of  places  in  a  cube 
and  its  root.  This  relation  is  exhibited  in  the  following  examples  : 

Cubes. 
1 

1,000 
1,000,000 
1,000,000,000 

Thus  we  perceive  that  a  number  consisting  of  one  place,  may 
have  from  one  to  three  places  in  its  cube;  and  that  in  all  cases  the 
addition  of  one  place  to  the  root  adds  three  places  to  the  cube. 
Hence, 

]f  a  number  le  pointed  off  into  three-figure  periods,  commencing 
at  units  place,  the  number  of  periods  icill  indicate  the  numb**  of 
places  in  the  root. 

341.  To  ascertain  how  the  several  figures  of  the  root  arc  related 
in  local  value  to  the  periods  of   the  power,  we  may  decompose  any 
number,  as  5423,  and  form  the  cubes  of  its  several  parts,  as  follows : 
50003  =  125  000  000  000 
5400s  =  157  464  000  000 
54203  =  159  220  088  000 
54233  =  159  484  621  967.  Hence, 


CUBE    ROOT    OF    NUMBERS.  177 

The  cube  of  the  first  figure  of  the  root  is  contained  wholly  in  the 
first  period  of  the  power  j  the  cube  of  the  first  two  figures  of  the 
root  is  contained  wholly  in  the  first  two  periods  of  the  p  wer  ;  and 
so  on. 

242.  To  employ  the  binomial  cube  as  a  formula  for  extracting 
the  cube  root  of  a  number,  we  must  represent  the  first  figure  or 
figures  of  the  root,  taken  with  their  local  value,  by  a,  and  the  re- 
maining figures  by  b.  The  operation  will  then  be  the  same,  in  form 
and  principle,  as  that  employed  in  extracting  the  cube  root  of  alge- 
braic quantities. 

1.  Let  it  be  required  to  find  the  cube  root  of  164,206,490,176. 

OPERATION. 

1642064901 76|5476 
125 


154 

616 

7500    39206 
8116    32464 

1627 

11389 

874800   6742490 
886189   6203323 

16416 

98496 

89762700  539167176 
89861196  539167176 

There  are  four  periods  in  the  given  number,  indicating  that  there 
will  be  four  figures  in  the  root.  As  the  cube  of  the  first  figure  will 
be  contained  wholly  in  the  first  period,  (241),  we  seek  the  greatest 
perfect  cube  in  164.  This  we  find  to  be  125  ;  its  root  is  5,  which 
we  write  as  the  first  figure  of  the  root  sought. 

We  may  now  consider  the  5  as  tens  of  the  next  inferior  order  in 
the  root,  and  let  a  =  50,  and  b  represent  the  next  figure.  A.nd  since 
the  cube  of  a-\-b  will  be  contained  wholly  in  the  first  two  periods  of 
the  number.  (241),  we  subtract  a3,  or  125.  from  164,  and  to  the 
remainder  bring  down  the  next  period,  making  39206.  Then  this 
result  must  contain  at  least  3aa6-f-3a6s-}-&s,  (236),  and  we  there- 
fore divide  it  by  3  a9,  or  7500,  as  a  trial  divisor,  and  obtain  4  for 
the  value  of  1>.  or  the  second  figure  of  the  root. 

To  complete  the  divisor,  we  have  3a-\-b  =  154  for  the  first  factor 
of  the  correction,  and  (3a-f- 6)6  =  616  for  the  first  correction  ; 

M 


178  EVOLUTION. 

whence  by  addition  we  obtain  8116,  the  complete  divisor.  Multiply- 
ing this  by  4,  and  subtracting  the  product  from  the  dividend,  we  have, 
after  bringing  down  the  next  period,  6742490  for  a  new  dividend. 

We  may  now  form  a  new  trial  divisor,  according  to  (388,  I). 
We  shall  have  8116-f  616-J-16  =  8748  j  or  874800,  if  we  give  to 
the  figures  their  local  value  with  respect  to  the  lowest  order  in  the 
dividend.  By  division,  we  have  7  for  the  next  figure  of  the  root. 
To  find  a  correction  for  the  new  trial  divisor,  we  annex  the  last 
figure,  7,  to  3  times  the  former  figures  of  the  root,  and  obtain  1627 
for  the  first  factor  j  and  we  then  continue  the  operation,  repeating 
the  former  steps,  till  the  work  is  finished. 

Hence  we  have  the  following 

RULE.  I.  Point  off  the  given  number  into  periods  of  three  figures 
each,  counting  from  units'  place  toward  the  left  and  right. 

IT.  Find  the  greatest  cube  in  the  left  hand  period,  and  place  its 
root  for  the  first  figure  of  the  required,  root ;  subtract  this  cube 
from  the  first  period,  and  to  tlie  remainder  bring  down  the  next 
period  for  a  dividend. 

III.  At  the  left  of  the  dividend  write  three  times  the  square  of  the 
root  already  found,  and  annex  two  ciphers,  for  a  trial  divisor  ;  divide 
the  dividend,  and  write  the  quotient  for  the  next  figure  of  the  root. 

IV.  To  three  times  the  first  figure  of  the  root  annex  the  last; 
multiply   this  result  by  the  last  root  figure,  for  a  correction  to  the 
trial  divisor;  add  the  correction,  and  the  result  will  be  the  complete 
divisor. 

V.  Multiply  the  complete  divisor  by  the  last  figure  of  the  root, 
subtract  the  product  from  the  dividend ',   and  to  the  remainder  bring 
down  another  period  for  a,  new  dividend. 

VI.  Add  together    the   last  complete  divisor,  the  last  correction, 
awl  the  square  of  the  last  figure  of  the  root,  and*annex  two  ciphers, 
for  a  new  trial  divisor  ;  then  l>y  division  obtain  another  figure  of 
the  root. 

VII.  Take  the  first  factor  of  the  last  correction,  multiplying  its 
right  hand  figure  by  3,  and  annex  the  last  figure  of  the  root,  for 
the  first  /actor  of  the  new  correction  ;  with  which  proceed  as  in  the 
former  steps,  till  the  work  is  finished. 


CUBE    ROOT    OF    i\ UMBERS. 


179 


EXAMPLES  FOR  PRACTICE. 


1.  Find  the  cube  root  of  148877. 

Ans.  53. 

2.  Find  the  cube  root  of  571787. 

Ant.  83. 

3.  Find  the  cube  root  of  256047875. 

Ans.  635. 

4.  Find  the  cube  root  of  354894912.  ' 

Ans.  708. 

5.  Find  the  cube  root  of  11852.352. 

Ans.  22.8. 

G.  Find  the  cube  root  of  144125083907. 

Ans.  5243. 

7.  Find  the  cube  root  of  128100283921. 

Ans.  5041. 

8.  Find  the  cube  root  of  105555,569176. 

Ans.  4726. 

9.  Find  the  cube  root  of  731189187729. 

Ans.  9009. 

10.  Find  the  cube  root  of  1762.790912. 

Ans.  12.08. 

11.  Find  the  cube  root  of  1061520150G01. 

Ant.  10201. 

12.  Find  the  cube  root  of  33212361:641984. 

Ans.  321.44. 

13.  Find  the  cube  root  of  1371737997260631 

.    Ans.  111111. 

14.  Find  the  cube  root  of  .171467. 

Ans.  .55555-f. 

15.  Find  the  cube  root  of  .004235801032. 

Ans.  .1618. 

CONTRACTED    METHOD. 

243.  In  applying  the  method  of  contracted  decimal  division  to 
the  extraction  of  the  cube  root  of  a  number,  we  observe, 

1st.  For  each  new  figure  in  the  root,  the  terms  in  the  operation 
extend  to  the  right  3  places  in  the  column  of  dividends,  2  places  in 
the  column  of  divisors,  and  1  place  in  the  extreme  left-hand  column- 
Hence, 

2d.  If  at  any  point  in  the  operation  we  omit  to  bring  down  new 
periods  in  the  dividend,  we  must  shorten  each  succeeding  divisor  1 
place,  and  each  succeeding  term  in  the  left-hand  column  2  places. 

3d.  If,  however,  for  the  -first  contraction  in  the  column  of  divisors, 
and  in  the  left  hand  column,  we  simply  omit  the  ea-tcinlcd part^  and 
afterward  contract  according  to  the  precept  just  given,  each  con- 
tracted multiplicand  will  have  one  redundant  figure. 


180  EVOLUTION. 

1.  Find  the  cube  root  of  850  correct  to  8  decimal  places. 

OPERATION. 

[9.472682374-,  root. 
850.000000  ~ 
729 


274-   1096 

24300  121000 
25396  101584 

2327  19789 

2650800  19416000 
2070589  18694123 

2841   568 

2690427   721877* 
2690995   538199 

28     17 

269156    183678 
269173    161504 

26919     22174 
21535 

2692        639 
538 

269        101 
81 

27          20 
19 

We  proceed  in  the  usual  manner  until  we  reach  the  first  contract- 
ed dividend,  721877,  which  is  obtained  in  the  common  way,  the 
period  of  ciphers  being  omitted.  The  corresponding  trial  divisor, 
with  the  ciphers  at  the  right  omitted,  is  2690427,  the  right  hand 
figure  of  which  is  redundant,  being  of  an  order  lower  than  is 
required  to  obtain  a  product  corresponding  in  local  value  to  the 
contracted  dividend.  By  division,  we  have  2  for  a  new  figure  in 
the  root.  To  obtain  a  correction  whose  lowest  figure  shall  be  of 
the  same  order  as  the  lowest  in  the  trial  divisor,  we  form  the  term 
2841  in  the  common  way,  but  omit  to  annex  2,  the  Ir.st  figure  in 
the  root.  Then  2841 X 2  =  5682,  of  which  568  is  the  part  required 
for  the  correction.  We  then  have  2090995  for  a  complete  divisor, 
538199  for  a  product,  and  183678  for  the  new  dividend.  For  the 
next  trial  divisor,  we  add  2690995  and  568,  and  reject  one  figure, 
thus  obtaining  269156.  The  square  of  2,  the  last  root  figure,  is  of 


CUBE    HOOT    OF    NUMBERS.  181 

course  rejected,  on  account  of  its  inferior  local  value.  The  remain- 
ing part  of  the  operation  requires  no  further  explanation. 

It  will  be  seen  that  the  number  of  places  obtained  in  the  root  is 
equal  to  the  number  of  places  assumed  in  the  power.  Hence  wo 
have  the  following 

RULE.  I.  Assume  as  many  places  in  tJie  "power  as  there  are 
places  required  in  the  root,  and  proceed  in  the  usual  manner  till  all 
the  assumed  figures  have  been  brought  doicn. 

II.  Form  the  next  trial  divisor  as  usual,  omitting  the  ciphers  at  the 
right;  and  reject  one  place  in  forming  each  subsequent  trial  divisor. 

III.  In  completing  the  first  contracted  divisor,  omit  to  annex  the 
new  figure  of  the  root  to  the  corresponding   term  in  the  left-hand 
column,  and  reject  two  places  in  funning  each  succeeding  term  in. 
this  column. 

IV.  In  multiplying,  treat  the  right-hand  figure  of  each  contract- 
ed term  as  redundant,  both  in  the  column  at  the  left,  and  in  the  col- 
umn of  divisors. 

NOTE. — To  avoid  complicating  the  process  of  contracting,  it  is  better  to 
use  none  but  full  periods,  even  if  the  root  is  thereby  carried  beyond  the 
required  number  of  places. 

EXAMPLES    FOB  PRACTICE. 

1.  Find  the  cube  root  of  3  correct  to  6  decimal  places. 

Ans.  1.442249-f . 

2.  Find  the  cube  root  of  7  correct  to  6  decimal  places. 

Ans.   1.912931-f . 

3.  Find  the  cube  root  of  156  correct  to  8  decimal  places. 

Ans.  5.38321261+. 

4.  Find  the  cube  root  of  34786  correct  to  6  decimal  places. 

Ans.  32.643859-f . 

5.  Find  the  cube  root  of  10.973937  correct  to  6  decimal  places. 

Ans.  2.222222-f-. 

6.  Find  the  cube  root  of  1500.101520125  correct  to  8  decimal 
places.  Ans.  11.44740066-}-. 

7.  Find  the  cube  root  of  1.164132  correct  to  6  decimal  places. 

Ana.  1.051963 -r 
16 


182  RADICAL   QUANTITIES. 


SECTION    IV. 

RADICAL    QUANTITIES. 

344.  A  Eadical  Quantity  is  a  root  merely  indicated,  cither  by 
the  radical  sign  or  by  a  fractional  exponent  ;  as  3|/a,  $  'a  —  6, 

c(a-f  &)*,  m  ^x1  —  y*.    A  radical  quantity  may  be  either  surd  or 
rational. 

The  quantity  or  factor  placed  before  a  radical  is  its  coefficient. 
Thus  in  the  examples  just  given,  3,  1,  c,  andm  are  the  coefficients 
of  the  radicals. 

24«>.  The  Degree  of  a  radical  quantity  is  denoted  by  the  radical 

index,  or  by  the  denominator  of  the  fractional  exponent.     Thus, 

i 
I/a,  (a  —  6)2  are  radicals  of  the  2d  degree  ; 

~^x*  —  y,   a*l*  are  radicals  of  the  3d  degree; 

vac,    (x-f-y)»  are  radicals  of  the  nth  degree. 
24  6.  Similar  Radicals  are  those  in  which  the  same  quantity 
is  affected  by  radical  signs  having  the  same  index.     Thus, 

4  vV-fi,  —  ftf+b,  and  7  (aa-f-&)^  are  similar  radicals. 


REDUCTION  OF  RADICALS. 
CASE  I. 

947.  To  reduce  a  radical  to  its  simplest  form. 

A  radical  is  in  its  simplest  form  when  it  contains  no  perfect 
power  corresponding  to  the  degree  of  the  radical. 

1.  Reduce  l/48a"x*  to  its  simplest  form. 

We  have  seen  that  the  nth  root  of  a  quantity  is  equal  to  the  prod- 
uct of  the  nth  roots  of  its  component  factors;  (227).  Hence 
we  have 


REDUCTION.  188 

It  will  be  seen  that  we  first  separate  the  quantity  under  the  rad- 
ical sign  into  two  factors,  one  of  which  is  a  perfect  square.  Then 
according  to  the  principle  of  evolution  just  adduced,  we  have  the 
product  of  two  radicals,  one  of  which,  1/lGaV2,  is  rational,  and  the 
other,  l/3.r,  is  a  surd.  The  expression  is  then  reduced  by  extract- 
ing the  root  of  the  rational  part,  and  multiplying  it  by  the  surd. 


2.  Reduce  3F&*y  —  Sx*t/4  to  its  simplest  form. 
Factoring  as  before,  we  have 

Y  =  s  x  i^y  x 

3  X  2ry  X 


Hence  the  following 

RULE.  I.  Separate  the  factors  of  the  quantity  under  the  radical 
sign  into  two  groups,  one  of  which  shall  contain  all  tlie  perfect  pow- 
ers corresponding  in  degree  to  the  radical. 

II.  Extract  the  root  of  the  rational  part,  multiply  this  root  by 
the  given  coefficient,  and  prefix  the  product  to  the  surd  or  irrational 
part. 

EXAMPLES  FOR  PRACTICE. 

Reduce  the  following  radicals  to  their  simplest  form  : 


184  RADICAL    QUANTITES. 


13.  (2a'i5 — 3a6i7)5.  Ans.  a&(2a*— 31*] 

15.  l/8a2"x4'". 

16.  if  cTV^. 


17.  (2x9w 

18.  a-mc(amVn—  a^V1)*".  ^ns.  ca(cn—  amn)~n". 

248.  When  the  quantity  under  the  radical  sign  is  a  fraction, 
we  may  transform  it  in  such  a  manner  that  the  denominator  shall  be 
a  perfect  power  corresponding  in  degree  to  the  indicated  root. 
Then  after  simplifying,  the  quantity  remaining  under  the  radical 
sign  will  be  entire.  It  will  generally  be  expedient  to  separate  the 
given  fraction  into  two  factors,  one  of  which  shall  be  a  perfect 
power  }  we  may  then  operate  upon  the  surd  part  separately. 

1.  Reduce  1/4  i  to  its  simplest  form. 

OPERATION. 

l/4f  =  T/^XV  =  l/,ftxV  =  VAX  4X33  =  -&  1/33,  An*. 
2.  Reduce  v  -fa  to  its  simplest  form. 

OPERATION. 


In  like  manner  reduce  the  following  : 

3.  ^-i.  Ans. 

4. 

5. 


G. 

:  Ti- 


REDUCTION.        .  185 


CASE    II. 

249.  To  reduce  a  rational  quantity  to  a  radical,  or  to 

introduce  a  coefficient  of  a  radical  under  the  radical  sign. 

Since  involution  and  evolution  are  the  converse  of  each  other,  we 

have 

a  =  ytf  =  fa*  =  Va4,  etc. 

Whence,  we  have  also, 

ayb  =  i/a*Xi/b  =  V~c?b. 

We  have,  therefore,  the  following 

RULE.  I.  To  reduce  a  rational  quantity  to  a  radical  :  —  Involve 
it  to  a  power  denoted  by  the  degree  of  the  required  radical,  and  write 
the  result  under  the  radical  sign. 

II.  To  introduce  the  coefficient  of  a  radical  quantity  under  tho 
radical  sign  :  —  Involve  it  to  a  power  denoted  ly  the  degree  of  the  rad- 
ical f  and  multiply  the  quantity  under  the  radical  by  the  power  thus 
obtained. 

EXAMPLES    FOB  PRACTICE. 

1.  Reduce  ab*  to  a  radical  of  the  second  degree.    An*.  \/a*b*. 

2.  Reduce  5aa.ry3  to  a  radical  of  the  3d  degree. 

AM.  ^/125(t«.ry. 

3.  Reduce  a  —  cz  to  a  radical  of  the  4th  degree. 

Ans.  (a4—  4af«+6aW—  4<*cV+cV)i 

Introduce  the  coefficients  of  the  following  radicals  under  the  rad- 
ical sign  : 

4.  4ul/27yT  Ans.  1/32^.7^" 

5.  3.t2  Vx—  y.  Ans.  TV27x7 


6.  («_2i)1/2a,  Ans.  l/2as— 


16* 


186  RADICAL   QUANTITIES. 

CASE   III. 

2oO.  To  reduce  radicals  to  a  common  index. 

m  mr 

It  may  be  shown  that  an  =  a™,  r  being  any  integer  -whatever. 

Let  x  =  an  ;  (1) 

involving  (1)  to  the  «th  power,  xn  =  am,  (2) 

«       (2)     «       rth      "  af=a'",  (3) 

mr 

taking  the  nrth  root  of  (3),  x  =  a~,  (4) 

m  mr 

equating  values  of  x  in  (1)  and  (4),  an  =  anr  .     Hence, 

1  —  If  Loth  terms  of  a  fractional  exponent  be  multiplied  or  di- 

vided by  the  same  number,  the  value  of  the  expression  will  not  be 

altered. 

From  (233)  we  have 


and  d^ssrVa-r.  Hence, 

2.  —  If  both  the  index  of  a  radical  and  the  exponent  of  the  quan- 
tity under  the  radical  sign  be  multiplied  or  divided  by  the  same 
number,  the  value  of  the  expression  will  not  be  altered. 

1.  Reduce  (ab)"  and  (aax)     to  a  common  index. 


(a&)*  =  (a6)*  =  (af&')*l 

i  2  i  fAns. 

(aVp  =  (aV)S  =  (aV)«  J 


2.  Reduce  "^a'c  and  VxV  to  a  common  index. 


V»...,  V35- 

Ilence,  we  have  the  following 

RULE.  I.  When  the  quantities  are  affected  by  fractional  expo- 
rents  :  —  Reduce  the  yiven  exponents  to  their  least  common  denomi- 
nator ;  then  raise  each  quantitifto  a  power  denoted  by  the  numera- 
tor of  its  new  exponent,  and  affect  each  result  with  a  fractional 
index  equal  to  the  reciprocal  of  the  common  denominator. 


ADDITION.  187 

II.  When  the  quantities  are  affected  by  radical  signs  : — Find  the 
least  common  multiple  of  the  given  indices  for  the  common  index 
required  ;  and  raise  the  quantity  under  each  radical  sign  to  a  pow* 
er  indicated  Ly  tJie  quotient  of  the  new  index  divided  l)ij  the  given 
index. 


EXAMPLES  TOR  PRACTICE. 

1.  Reduce  a2,  (cd)  ^,and  (a'c)*  to  a  common  index. 

Ant.  a^,  (c«d*)  A,  (aV)A. 

2.  Reduce  (3a'x)^,  (2ax*)7,and  (5aV)&  to  a  common  index. 

AM.  (81aV)T2,  (8aV)T2,  (25aV°)T2. 

1  2 

3.  Reduce  (a  —  &)2  and  (a-J-i)3  to  a  common  index. 

,  or  (a1—  3a2 


Am. 

,  or  (a4-f  4a' 


4.  Reduce  a,  1/ac,  v  a*x,  and  ^2ac*  to  a  common  index. 


5.  Reduce  -j/2,  V2,  and  v/2  to  a  common  index. 

Am.  VlUa,  V82,  VTO. 
C.  Reduce  a*,  V  5x,  K^2ax,  and   ^4a^x  to  a  common  index. 


7.  Reduce  v  x  —  y,  v  x-\-y^  and   v  x2  —  ^a  to  a  common  index. 


8.  Reduce  V  ax,  *Vxy,  and  ^  ex  to  a  common  index. 

*t±.TlS*  Cb      •&      •  v      } 

ADDITION  OF  RADICALS. 

£51.  When  the  quantities  to  be  added  are  similar  radicals,  it  is 
evident  that  the  common  radical  part  may  be  made  the  unit  of  addi- 
tion ;  the  result  will  then  be  a  single  radical  whose  coefficient  is  the 
sum  of  the  coefficients  of  the  given  radicals.  Radicals  which  do 


188  KADICAL    QUANTITIES. 

not  appear  to  be  similar,  may  become  similar  when  reduced  to  their 
simplest  forms. 

1.  What  is  the  sum  of  71/oe,  Sl/oc,  and  51/oc? 

==15l/^r    Ant. 

2.  What  is  the  sum  of  1^8^  1^27Vc,  and 

OPERATION. 


Sum  =  (5a-|-4c)^/aac, 

If  the  given  radicals  are  dissimilar,  the  addition  can  only  be  indi- 
cated. Hence  the  following 

RULE.  I.  Reduce  each  radical  to  its  simplest  form. 

II.  If  the  resulting  radicals  are  similar,  add  their  coefficients, 
and  to  the  sum  annex  the  common  radical ;  if  dissimilar,  indicate 
the  addition  by  the  proper  signs. 

EXAMPLES   FOR  PRACTICE. 

1.  Find  the  sum  of  ViQa*x  and  l/4aax.  -4ns.  Qa^/x. 

2.  Find  the  sum  of  1/32^  Vll,  and  1/128.  Ans.  18]/2. 

3.  Find  the  sum  of  &W,  V f35,  and  1^625.  Ans.  10^/5. 

4.  Find  the  sum  of  1^108,  9^4,  and  1^1372.  Ans.  19^4. 

5.  Find  the  sum  of  -j/4,  i/f,  and  i/yg.  Ans.  -j/2. 

6.  Find  the  sum  of  f^b  ^IJ,  an  1  TV|f|.  Ans.  |J^3. 

8  ~ 


7.  Find  the  sum  of  j^J,  |1/^  and  Jl/j  J.       ^ln«.  |-J|/'G. 

8.  Find  the  sum  of  Sl/abm*,  wl/4o6,  and  V25abni\ 

Ans.  10ml/«6. 

9.  Find  the  sum  of  2aT/c2x-  —  c2^,  3cl/aaic—  a>,  and  • 
5l/aV.r—  a'c'.  J.TW.  lOacl/  a^y 


SUBTRACTION.  189 

10.  Find  the  sum  of  l/20afw— 2Uoc7»-f  5mc»  and 
l/20ni£a — 60ac»i-j-45a*»i.  Ans.  (c — a 

11.  Find  the  sum  of  S^'a?,  ^a?,  and  2^xd 

-An*. 

12.  Find  the  sum  of  5a(cx*—dx*^  and  2* 

Ans.  %ax(c — cQ3. 

13.  Find  the  sum  of  \    ^~  '  >    \f '  ^,    \   and 

'     a-j-6          *    a — 6 

.  21/aa— 6»- 


14.  Find   the    sum    of    V(l+ a)'1,    I/a2  (1 -fa)-1,    and 

<>&=SF>.  '    VWa 


SUBTRACTION    OF    RADICALS. 

.  When  the  radicals  are  similar,  it  is  evident  that  we  may 
make  the  common  radical  the  unit  of  subtraction.  Hence  the 
following 

RULE.  I.  Reduce  each  radical  to  its  simplest  form. 

II.  If  the  resulting  radicals  are  similar •,  find  the  difference  of 
the  coefficients,  and  to  the  result  annex  the  common  radical  part  j 
if  dissimilar,  indicate  the  subtraction  by  the  proper  sign. 

-    EXAMPLES    FOB    PRACTICE. 

1.  From  41/135  take  21/60L  Ans.  8l/lK 

2.  From  1/75  take  1/50.  Ans.  5(|/3— !/2). 

3.  From  3l/16^>  take  3l/^T  Ans.  (12a9 


4    From  \y  ~  •'.?  take  '  P'  STT  3  Ans    i-i/11 

8  1^5 

4 


2      /490a« 
888 


190  RADICAL   QUANTITIES. 

6.  From  (aV—  3c'x°)*  take  2(aV—  3dax^. 

Am.  (c—  2d)  (a* 

7.  From  (a1—  a&'+a'fc—  &•)*  take  (a'—  3a'&+3a6a—  £')*. 


8.  From  o\/-  take  6\/-  ~n—    ^-ns-    --  fl/a  —  1. 

X  -  1  *      X-j-1  X3  -  1 


MULTIPLICATION  OF  RADICALS. 

.  It  has  already  been  shown  (S3  7)  that  the  wth  root 
of  the  product  of  two  or  more  factors  is  equal  to  the  product  of 
the  nth  roots  of  those  factors.  And  since  the  converse  of  this  pro- 
position is  true,  we  shall  have 

Va  X  V&  =  V  ob- 

it' the  radicals  have  coefficients,  the  product  of  the  coefficients 
may  be  taken  separately.     Thus, 


IF  the   radicals   have    not   a  common  index,  they  must  first  be 
reduced  to  the  same  decree. 

O 

Let  it  be  required  to  find  the  product  of  a^/x  and 


OPERATION. 

ay'x  =  a  \ 


Product,  ab       y  = 
Hence  the  following 

RULE.  I.  If  necessary,  reduce  the  given  radicals  to  a  common 
indrx. 

II.  Multiply  the  quantities  in  the  radical  parts  together,  and  place 
Jie  product  under  the  common  radical  sign  ;  to  this  result  prefix  the 
product  of  the  given  coefficients,  and  reduce  the  whole  to  its  simplest 
form. 


DIVISION.^.  £>>  1 91 


EXAMPLES  FOR  PRACTICE. 

Find  the  following  indicated  products: 

1.  5/5x3/8.  Ans.  301/10. 

2.  41/I2X3/2.  Ans.  24/6. 

3.  3/2X2/8.  Ans.  24. 

4.  2/5x2l/10x3/6.  Ans.  120/3. 

5.  21^4x3/4.  Ans.  121 

6.  5c 
7. 


9.  i^ISXl/ia  ^ws.  V225000 

a  lx    y  s  /^2    3 1^5 

10.   -7J-  X- -\KX\f~i   •  Ans. 

Z>\y       x  \a2        V/j/9 


OF  RADICALS. 

2«><J:.  Since  a  fraction  is  raised  to  any  power  by  involving  its 
numerator  and  denominator  separately  to  the  required  power,  it  is 
evident  that  any  root  of  a  fraction  will  be  obtained  by  extracting  tho 
required  root  of  each  term  separately.  Hence  we  have 


Conversely,  we  shall  have 

a  j 


The  quotient  of  the  nth  roots  of  two   quantities  is  equal  to   the  nth 
root  of  their  quotient. 

Upon  this  principle  is  based  the  rule  for  the  division  of  radicals. 


102  EADICAL   QUANTITES. 

1.  Divide  6a9VTc  by  Sa^c. 

GaVZTc       Ga«    fUc 

-  =  K—  \/-  =  2ai/i,  Ans. 
oc<i/c        3a   *c 

2.  Divide  ^  '  x*y  by  I/  a*/. 


Hence  the  following 

RULE.  I.  If  necessary,  reduce  the  radicals  to  a  common  index. 

II.  Divide  the  coefficient  of  the  dividend  by  the  coefficient  of  the 
divisor  ;  divide  also  the  quantity  in  the  radical  part  of  the  dividend 
by  the  quantity  in  the  radical  part  of  the  divisor,  placing  ^ie  27M?- 
tient  under  the  common  radical  sign.  Prefix  the  former  quotient 
to  the  latter,  and  reduce  the  result  to  its  simplest  form. 


EXAMPLES  FOR  PRACTICE. 

1.  Divide  4l/50~by  2j/5.  Ans.  2l/I(X 

2.  Divide  6^100  by  3^5.  -4ns.  2^W. 

fl/ / A         6  /  10tt 

3.  Divide  V  20aaJ  by  l/15ai/.  -*ns-  \/         • 

*  IbOC^ 

4.  Divide  (a'^73)^  byc?i.  J.;is.  (ai)i 

5.  Divide  (16a3— 12a*x)^  by  2a.  ^Ins.  (4a— 3x)i 

6.  Divide  45  by  S^S.  ^Iws.  3^5. 

7.  Divide  (o6V)i  by  (a'iV)i  ^Ins.    "4/Z!. 

^«c* 

8.  Divide  12c*(a — x)$  by  4c(a— x)^.  ^4ns.  3c(a — x)T^. 

—  —  _i 

9.  Divide  (asc)m  by  (ac8)"  .  .4ns.  (a^-^c"-*"1)"*. 

10.  Divide  ^-  by  TTT-  -4.1'-*.  y -"•• 

11.  Divide    Va*b—ab*  by  T/S. 


POWERS    AND    ROOTS.  193 

POWERS  AND  ROOTS  OF  RADICAL  QUANTITIES. 

.  According  to  the  rule  for  multiplication  of  radicals,  to 
j_ 

form  the  ??ith  power  of  aw  ,  or  V«,  we  must  take  the  quantity,  a, 
m  times  as  a  factor,  and  affect  the  result  by  the  common  radical 
index.  Hence, 

(«V  =  «T~  =  a% 

or  (V a)"*  =  Vam.  That  is, 

The  mth  power  of  the  nth  root  of  a  quantity  is  equal  to  the.  nth 

root  of  the  mth  power  of  that  quantity. 

\_ 

256.  To  obtain  the  with  root  of   the  radical  a* ,  or  V«,  wo 
may  proceed  as  follows :     Let 

-  -          / 

x  =  (a* )-  01  r/v*  a) 

Involving  both  members  of  (1)  to  the  mth  power, 

i_ 
xm  =  a"  or  V«J  (2) 

involving  both  members  of  (2)  to  the  nth  power, 

x"**  =  a  or  a  ;  (3) 

taking  the  wnth  root  of  each  member  of  (3), 

i_ 
x  =  a*"1  or  mty/a'}  (4) 

hence  by  equating  values  of  x  in  (1)  and  (4), 

_L  _L          JL 
(«")»  =  a", 


or, 


"K/a:="Va.  That  is, 


TJie  mth  root  of  the  nth  root  of  a  quantity  is  equal  to  the  mnth 
root  of  that  quantify. 

'.  Since   r/V«  =  mV«,  and    *  j"(/a  —  "V^,  we  have 

That  is, 


\ 

nth  root  of  a  quantity  is  the  same  as  the  nth 
root  of  the  mth  root  of  that  quantity. 

17  N    " 


194  RADICAL    QUANTITIES. 

CASE  I. 

258.  To  involve  a  radical  quantity  to  any  power. 

]  .  Raise  (ab)*  to  the  2d  power. 
From  (255)  we  have 


-'.  Raise  ^2ax  to  the  4th  power.  ' 
From  (255)  we  have 


But  since  6  =  2x3,  we  have,  from  (256), 


6v/16aV  = 

In  practice,  the  simplification  may  be  effected  by  canceling  a 
factor  from  the  index  of  the  radical,  and  extracting  the  correspond- 
ing root  of  the  quantity  under  the  radical  sign.  Thus,  in  general 
terms,  we  have 


Hence  the  following 

RULE.  I.  If  the  quantity  is  affected  by  a  fractional  exponent, 
multiply  this  exponent  by  the  index  of  the  required  power. 

II.  If  the  quantity  is  affected  by  the  radical  sign,  raise  the  quan- 
tity under  the  radical  sign  to  the  required  power  ;  and  if  the  result 
is  a  perfect  power,  of  a  degree  corresponding  to  any  factor  of  the 
radical  index,  cancel  this  factor  from  the  index,  and  extract  the 
corresponding  root  of  the  quantity  under  the  radical  si  ;n. 

NOTE.  —  The  coefficient  may  be  involved  separately. 

EXAMPLES    FOR    PRACTICE. 

1.  Raise  V~2a*  to  the  3d  power.  Ans.  a^/Sa. 

2.  Raise  tfx*y*  to  the  2d  power.  Ans.  xyfxy. 
8.  Raise  3^4^  to  the  4th  power.  Ans.  lQ2a.9^2a^~ 
4.  Raise  (a  —  &)*  to  the  2d  power.  Ans.  (a  —  Vf*, 


POWERS    AND    ROOTS.  195 

5.  Raise  Vl2a&9  to  the  5th  power.  Ans.  26l/3a. 

6.  Raise  ^c(a — x)9  to  the  8th  power.  

Ans.  (a — x)  V  ca(a  — x). 

7.  Raise  axl/ax  to  the  4th  power.  Ans.  a*x*. 

8.  Raise  "Px'y* — x2y8  to  the  2d  power.  Ans.  xy^xy(x — yf. 

9.  Raise  (a-f-x)"5  to  the  6th  power.          Ans.    a* -\-2ax-\-x*. 

d  10 . 2of* 

10.  Raise  -  ^96cxe  to  the  2d  power.  AM.    — 

CASE  n. 
259.  To  extract  any  root  of  a  radical  quantity. 


1.  What  is  the  square  root  of 

Since  the  coefficient  is  a  perfect  square, 


But  from  (257)  we  have 


2.  What  is  the  6th  root  of  5ccfl/5c? 

Passing  the  coefficient  under  the  radical  sign,  we  have 

5cd3l/57=  l/125c'e/'. 
But  by  (»56), 


Reducing  this  result  by  canceling  the  factor  3  from  the  radical 
index,  and  taking  the  cube  root  of  the  quantity,  we  haye 


,  Ans. 

3.  What  is  the  4th  root  of  (ac)l  ? 
By  (856)  we  have 

{(ac)^  }^  =  (ac)'  X  *  =  (ac) 
Hence  the  following 

RULE.  I.  If  the  quantity  is  affected  ~by  a,  fractional  exponent, 
divide  this  exponent  by  the  index  of  the  required  root. 


196  RADICAL    QUANTITIES. 

II.  If  the  quantity  is  affected  by   the  radical  sign,  extract  the 
required  root  of  the  quantity  under  the  radical  sign,  if  possible ; 
otherwise,  multiply  the  index   of  the  radical  by  the  index  of  the 
required  root,  and  simplify  the  result  as  in  Case  I. 

III.  If  the  given  radical  has  a  coefficient,  extract  its  root  sepa- 
rately when   possible;    otherwise,  pass   the   coefficients   under   the 
radical. 


EXAMPLES   FOE  PRACTICE. 

1.  Find  the  cube  root  of  21/^7 

2.  Find  the  cube  root  of  aVaW. 

3.  Find  the  4th  root  of  21^98. 

4.  Find  the  square  root  of  f  ^486. 

5.  Find  the  square  root  of  49a4V  abx. 

6.  Find  the  cube  root  of  5|/5. 

7.  Find  the  6th  root  of  (^  )*- 

V  cy  I 

8.  Find  the  4th  root  of  ty f  Ans. 
26O.  The  principle  established  in  (256),  viz.,  that 


may  be  conveniently  applied  to  the  extraction  of  the  higher  roots 
of  quantities,  when  the  index  of  the  required  root  is  a  composite 
number. 

EXAMPLES. 

1.  Required  the  4th  root  of  8603056. 

Since  4  =  2x2,  we  take  the  square  root  of  the  square  root  of 
the  given  number.     Thus, 

1/8603056  =  2916  ;  1/2916  =  54,  Ans. 

2.  Required  the  6th  root  of  117649. 
Since  6  ='2x3,  we  have 

1/117649  =  343  ;  V 343  =  7,  An*. 


THEORY   OF    EXPONENTS.  197 

3.  Required  the  4th  root  of  1296.  Ans.  6. 

4.  Required  the  6th  root  of  177978515625.  Ans.  75. 

5.  Required  the  6th  root  of  191102976.  Ans.  24. 

6.  Required  the  8th  root  of  65536.  Ans.  4. 

7.  Required  the  4th  root  of  a4— 8a'6-f-24aa&*— 32ai'-f  16&4. 

Ans.  a—2b. 

8.  Required  the  6th  root  of  a19-|-6a10&+15a8&9-f  20aV-|-15aV 
6a'6§--&'.  Ans.  a'6. 


GENERAL  THEORY  OF  EXPONENTS. 

261.  It  has  already  been  shown  that 

amX«n  =  aro+w,   ~  =  a"*""1,  and  (am)*  =  a"-, 

m  and  n  being  integers,  and  either   positive  or  negative.     To  prove 
that  the  above  relations  are  true  universally,  it  remains  only  to  show 
that  they  hold  true  when  m  and  n  are  fractional. 
We  will  therefore  place 

m  =  —  and  n  =  — 

0  * 

?        !:       *+!! 

I.  To  show  that  a9  x  a*  =a9     '  . 

Reducing  the  exponents  to  a  common  denominator,  we  havo 


But  from  the  nature  of  fractional  exponents,  (222),  the  sec- 
ond member  of  this  equation  may  be  written 


and  as  the  two  factors  have  the  same  radical  index   (227)  the 
result  reduces  to 


and  since  ps  and  qr  are  integral,  this  last  result  becomes 

J_  P*+V  1-1.- 

(or***)*.  =  a  v    =a«     '. 

17* 


198  KADICAL   QUANTITIES. 

Thatia, 


which  was  to  be  proved. 

*      L        *_*- 
H.  To  show  that  a'+ct  =  af    '. 

By  transformations  similar  to  those  just  employed, 


(O* 

-(5)"1 


^ 

=  a  «*   =  a«    •  , 

wliich  was  to  be  proved, 

€  -       ZL 
III.  To  show  that  («')'=  a*". 

Let  us  place 

x  =  (a«~)'.  W 

Involving  (1)  to  the  power  denoted  by  «, 

*•  =  (<)';  « 

by  (255)  equation  (2)  becomes 

x'  =  £;  3> 

involving  (3)  to  the  power  denoted  by  q, 

x*'  =  a";  (4) 

extracting  the  root  whose  index  is  qs, 


hence,  by  equating  values  of  x  in  (1)  and  (5), 

p~  -        m. 
(<*)'  =  a», 

which  was  to  be  proved. 

We  conclude,  therefore,  that  in  multiplication,  division,  involution 
and  evolution,  the  same  rule  will  apply,  whether  the  exponents  are 
positive  or  negative,  integral  or  fractional 


THEORY   OF    EXPONENTS.  199 

EXAMPLES. 
1221 

1.  Multiply  a~b*  by  a363,  and  simplify  the  product 

2.  Simplify  the  expression 


3.  Multiply  x4_3x2-j-      by      — 

OPERATION. 


—  3x4+9x^—3x1 


4.  Divide  x—  5Vx»—  5^xs—  5^/x—  6^x    by  ^xf— 


OPERATION. 


—  9|/x 


5.  Multiply  x^  by  x^.  ^n«.  x^. 

6.  Multiply  a'fti  by  aM.  J.ns.   a'l/ail 

7.  Find  the  product  of  a*,  a*,  a^,  and  a~*.  Ans.  T^a*. 

8.  Divide  aM  by  a^c^  ^4n5.    (-\  *. 


200  RADICAL   QUANTITIES. 

9.  Divide  (**)*  by  (*')*. 

10.  Multiply  a-— a4  by  a^-j-1.  Ans.  a'— a*. 

11.  Multiply  2^x8-f  l/^" by  3^*— T/ayl 

.4ns.  C.r  4-  0 Vaty3  —  QVafyt—xy. 

12.  Multiply  a*-— 2a~i-}-a-1  by  a*  — a~*. 

Ans.  a^— 3-}-3a~J — «~-. 

13.  Divide  a — Z>  by  i/a  +i/b.  Ans.  -|/a — i/b. 

14.  Divide  a^— 2a2-j-a^  by  a^  —1.  -4ns.  a^— a« 

15.  Multiply  J+a'^+a^-fai'-f  a^^-f  ^  by  a^— ^. 

Ans.  a'—i1. 

16.  Divide  ^-f^a^-fa^  by  a^-f-z^-f al 

^Ins.  xs — x^a^-^d3. 

17.  Simplify  (a^-a^)TT.  -4ns  a3. 


18.  Simplify     (— ^-H  •  ^««-  2±v/5. 

i         i     «• 

{KfWCO}' 

19.  Simplify     \ r^ 

20.  Si 


3^72-3(3)3 
21.  Simplify        -t-.         Jut.  |(2 


Cl/i3+3)(Vi3-f  1/3)(Vl3— 1/3)  J 

^jw. 


IMAGINARY    QUANTITIES.  201 

IMAGINARY  QUANTITIES. 

969.  It  has  been  shown  (998  ?  3)  that  an  even  root  of  a  nega- 
tive quantity  is  imaginary,  an  expression  for  such  a  root  being  a 
symbol  of  an  impossible  operation.     Thus  if  w«  take  aa,  which  is 
numerically  a  perfect  square,  and  affect  it  with  the  minus  sign,  we 
can  not  obtain  the  square  soot  of  the  result.     For  we  have 
(+a)«  =  -fa', 
(-ay  =  +  a'. 

Hence  the  indicated  root,  V  —  aa,  is  not  real  but  imaginary.  Such 
expressions  are,  however,  of  frequent  occurrence  in  analysis  and  its 
application  to  physical  science,  and  conclusions  of  the  highest  impor- 
tance depend  upon  their  use  and  proper  interpretation.  We  there- 
fore proceed  to  investigate  the  rules  to  be  observed  in  operating 
with  such  quantities.  • 

263*  When  a  real  and  an  imaginary  quantity  are  connected  in 
a  single  expression,  the  whole  is  considered  imaginary  on  account  of 
the  presence  of  the  imaginary  part.  Thus  the  binomial,  4-j-F  —  3, 
considered  as  a  single  quantity,  is  imaginary. 

964.  According  to  (927'),  we  may  have 

1/HTa  =  1/«X(—  1)  =  i/o  '  V—l  ; 

also,       V—  a*—  63+2aft  =  V  (a—  &)'X(—  1)  =  (a—  V)V~\. 
Hence,  if  we  regard  only  quadratic  expressions,  every  imaginary 
quantity  may  be  reduced  to  the  form, 


in  which  a  is  the  real  part,  b  the  coefficient  of  the  imaginary  part, 
and  v  —  1  the  imaginary  factor.  Thus  we  may  employ  only  the 
single  symbol,  V  —  1,  to  indicate  that  a  quantity  is  imaginary. 

965.  For  convenience  in  multiplication  and  division  of  imagi- 
nary quantities,  we  will  now  obtain  some  of  the  successive  powers 
of  the  symbol  v  —  1,  and  deduce  the  law  of  their  formation. 

=+v'=i, 

-i)'  =    o/-i)  x  o7-!)  ==  -i,  ; 

-i)'  =          (-1)  X  (V7-!)  =  -V~\, 

-!)4  =  (-V=i)  x  (i/—  1)  =  4-1, 


202  RADICAL   QUANTITIES. 

Multiplying  these  powers,  in  their  order,  by  the  4th,  we  shal  . 
obtain  the  5th,  6th,  7th  and  8th,  the  same  as  the  1st,  2d,  3d,  and 
4th  ;  and  so  on. 

2GG.  The  common  rules  for  multiplication  and  division  of 
radicals  will  apply  to  imaginary  quantities,  with  a  simple  modifica- 
tion respecting  the  law  of  signs. 

Let  it  be  required  to  find  the  product  of  I/  —  a  and  I/  —  b. 

To  obtain  the  true  result,  we  must  separate  the  imaginary  symbol 
V  —  1  from  each  factor.  Thus 

l/ITa  X  V~l)  ==  i/a  -1/—L  X  j/fc  •!/—  L 

=  Vai  x  <y~w 

=  Val>  X  (—  1)  [From 


a  real  quantity,  and  negative. 

But  if  we  multiply  by  the  common  rule  for  radicals,  (253),  we 
shall  have 


a  result  erroneous  with  respect  to  the  sign  before  the  radical. 
Proceeding  as  in  the  first  operation  we  find  that 

(_l/H^)x(— V~b)  =-  -f-T/o6  •  (—1)  =  — 1/ofc ; 

Thus,  like  signs  produce  — ,  and  unlike  signs  produce  -j-.     Hence, 
1. —  The  product  of  two  imaginary  terms  will  be  real,  and  the 
sign  before  the  radical  will  be  determined  by  the  common  rule  re- 
versed. 

We  may  operate  in  like  manner  in  division  of  imaginary  quanti- 
ties.    Thus, 

-\-V~ab  -  V—\ 


_T/_fffc        —Val  -  1/—  1 


That  is,  like  sipis  produce  -j-  and  unlike  signs  produce  —  .    Hence, 


IMAGINABY    QUANTITIES.  203 

2.  The  quotient  of  one  imaginary  term  divided  "by  another  will 
bf  real,  and  (lie  sign  before  the  radical  will  be  governed  by  the  com- 
mon rule. 

2G7.     Let  us  assume  the  equation 

a-f  bV—l  =  a'+b'V—l,  (1) 

in  which  a  and  a!  are  real.     By  transposition, 

a— a!  =  (bf—  &)!/— L  (2) 

Now  it  is  evident  that  in  this  equation 

a  =  a'. 

For,  if  a  >  o!  or  a  <a',  then  the  first  member  of  equation  (2)  is 
different  from  zero,  and  real.  But  this  can  not  be,  because  the  sec- 
ond member  is  either  nothing  or  imaginary.  Hence  a  =  a' ;  and 
equation  (2)  becomes 

0  =  (V—b)V—l, 
which  can  only  be  satisfied  by  putting 

b  =  V.  Hence, 

If  two  imaginary  quantities  are  equal,  then  the  real  parts  are 
equal,  and  the  coefficients  of  the  imaginary  symbol  are  also  equal. 
2GS.  These  principles  may  now  be  applied  in  the  following 

EXAMPLES. 

1.  Multiply  aV^c  by  bV^d.  Am.  —abV^d. 

2.  Multiply  21/^6  by  1/H15.  Am.  —61/10. 

3.  Multiply  —  V^a~c  by  V— ad.  Ans.  aV~cJ. 

4.  Multiply  3l/=2  by  v'b.  Ans.  31/^40. 

5.  Multiply  3+1/— 5  by  7— 1/~5.  Ans.  26+4V/~5. 

6.  Multiply  j/a  -fl/^c  by  V~a-\-^/c.  Ans.  (a+c)V~l. 

7.  Divide  91/HlO  by  31/~2.  Ans.  3^/5. 

8.  Divide  al/^Tby  cV^d.  Ans.  -A/~ 

c  \rf 


204  RADICAL    QUANTITIES. 

T/HT 

9.  Reduce  — —  -  to  simpler  terms.  Ans.  *j 

2V— 3 

10.  Expand  (a-|-T/~r)4. 

Ans.  a4— Ga'c-fc'-f  (4a8— 4ac}V~c. 

11.  Divide  a'-f  l/^i  by  a— V~a.          Ans.  a-f-l/^T— 1. 

12.  Find  the  values  of  a;  and  y  in  the  equation  a-\-y-\-xV — c 


\y  =  c-fVac. 

PROPERTIES  OF  QUADRATIC  SURDS. 

269.  A  Quadratic  Surd  is  the  square  root  of  an  imperfect 
square. 

270.  The  root  of  a  number  will  be  a  surd,  when  the  number 
contains  one  or  more  irrational  factors.      Thus  I/ 12  is  a  surd,  for 
Vi-  =  2-|/3.     The  surd  i'actor  -|/3,  is  called  the  irrational  part  of 
the  given  surd. 

271.  A  quantity  may  be  a  surd  when  considered  algebraically, 
even  though  its   numerical  value  is  rational.      Thus,  the  quap^ty* 
v  a~\-"li  is  a  surd,  considered  as  an   algebraic  expression. 

a  —  13  and   I  —  6,  we  have  l/a-f26  —  1/I3+T2"=  1/i      =  5, 
a  rational  quantity. 

272.  The  following  properties  of  surds  arc  important  both  in  a 
theoretical  and  a  practical  view.      The  radical  expressions  are  sup- 
posed to  represent  irrational  numbers. 

1. — The  product  of  two  quadratic  surds  which  have  not  the  same 
irrational  part,  is  irrational. 

Let  a^/b  and  c\/d  be  the  two  surds,  reduced  to  their  simplest 
form.  Their  product  will  be 

acVld. 

And  since,  by  hypothesis,  b  and  d  are  not  the  same  numbers, 
one  of  them  must  contain  at  least  a  factor  which  the  other  does 
not  put  this  factor  must  be  irrational,  otherwise  the  given  surds 


QUADRATIC    SUKDS.  205 

are  not  in  their  simplest  form.      Therefore  the  product  acl/ld,  is 
irrational;  (27  O). 

2- — The  Kum  or  difference  of  two  quadratic  surds  ichich  have  not 
the  same  irrational  part,  can  not  be  equal  to  a  rational  quantify. 

Let  |/a  and  ^/L  be  the  two  surds ;  and,  if  possible,  suppose 

l/a+!/6  =  c,  (1) 

c  being  rational.     Squaring  both  members,  and  transposing  a-{-b, 
2V 'ab  =  c'—a—b.  (2) 

That  is,  we  have  an  irrational  quantity  equal  to  a  rational  quan- 
tity, which  is  impossible.  Therefore  equation  (1)  cannot  be  true. 

In  like  manner  it  can  be  shown  that  the  difference  of  two  surds, 
not  having  the  same  irrational  part,  can  not  be  rational. 

3. —  The  sum  or  difference  of  two  quadratic  surds  which  have 
not  the  same  irrational  part,  can  not  be  equal  to  another  quadra  fie 
surd. 

If  possible,  suppose  j/a-f-|/&  =  i/c,  in  which  c  is  rational,  but 
l/c  a  surd. 

Squaring  both  members,  and  transposing, 

2V/~al  =  c—a—b, 

which  is  impossible,  because  a  surd  can  not  be  equal  to  a  rational 
quantity. 

4. — In  any  equation  which  involves  both  rationed  quantities  and 
quadratic  surds,  the  rational  parts  on  each  side  are  equal,  and  also 
the  irrational  parts. 

Suppose  we  have 

a+ftj/o;  =  c+rfj/y,  (1) 

the  surds  being  in  their  simplest  form.     By  transposition, 
^  fcjA— d^/y  ==  c—a.  (2) 

Since  the  second  member  is  rational,  equation  (2)  can  not  be  true 
if  the  surds  have  not  the  same  irrational  part;  (2).  Therefore 
1/x  =  i/y,  and  the  equation  may  be  written, 

(b— d)1/x  =  c— a,  (3) 

which  can  be  true  only  when  b — d  =  0  and  c — a  =  0  ;  for  other- 
wise, we  should  have  a  surd  equal  to  a  rational  quantity  or  to  zero: 
Hence,  in  (1),  a  ==  c,  and  b^/x 

18 


206  RADICAL    QUANTITIES. 

SQUARE   ROOT   OF   A   BINOMIAL   SURD. 

273.  A  Binomial  Surd  is  a  binomial,  one  or  both  of  whose 
terms  are  surds.     Thus,  3-f  j/5  and  j/7—  -j/2  are  binomial  surds. 
974.  If  we  square  a  binomial  surd  in  the  form  of  a  ±  ]/b  or 
I/a  db  \/b,  the  result  will  be  a  binomial  surd.     Thus, 

(S-fv/5)8  =  14+6J/5  ; 
(J/7--J/2)*  =  9—21/14: 

Hence,  a  binomial  surd  in  the  form  of  a  db  i/b  may  sometimes 
be  a  perfect  square. 

275.  To  obtain  a  rule  for  extracting   the  square  root  of  a 
binomial  surd  in  the  form  of  a  it  \/b,  let  us  assume 

!/*+!/*  =  l/^h/fc,  W 

in  which  one  or  both  of  the  terms  in  the  first  member  must  be 
irrational,  because  the  second  member  is  a  surd. 
Squaring  both  members, 


Hence  from  (279,  4)  we  have 

x+y  =  a,  (?) 

21/tty  =  !/6.  (4) 

Subtracting  (4)  from  (3),  and  then  taking  the  square  root  of  the  result, 

Vx  —  i/y  == 

Multiplying  (1)  by  (5), 

=  Ia1—  6. 
Combining  (3)  and  (6)  we  obtain 

~-°+*/^i==*,  (7) 


Now  it  is  obvious  from  these  equations  that  x  and  y  will  bo 
rational  when  a*  —  b  is  a  perfect  square.  Moreover,  the  values  of 
x  and  y  in  (7)  and  (8)  will  evidently  satisfy  equations  (1)  and  (5). 
Hence,  to  obtain  the  square  root  of  a  binomial  surd,  we  may  pro- 
ceed as  follows  : 


QUADRATIC    SURDS.  207 

Let  a  represent  the  rational  part,  and  -j/6  the  radical  part,  and 
find  the  values  of  x  and  y  in  equations  (7)  and  (8).  Then  if  the  bi- 
nomial is  in  the  form  of  a+-j/£,  as  in  equation  (1),  the  required 
root  will  be 


But  if  the  binomial  is  in  the  form  of  a  —  j/5,  as  in  equation  (5),  the 
required  root  will  be 


EXAMPLES  FOR  PRACTICE. 


1.  Required  the  square  root  of  7-|-4j/3. 
In  this  example  a  =  7,  and  j/6  =  4|/3  ;  or  6=48.     Hence, 

48" 

=4; 


And  wo  have 

T/aj+T/y  =  2+1/3,     An*. 

2.  Required  the  square  root  of  11  —  8  V  —  5. 

In  this  example  a  =  11  and  j/6  =  81/^5,  or  b  =  —320.     We 
have,  therefore, 

11+1/121+320 
a=-       -2-       -  =  16; 

11—1/121+320 


Hence,  I/a  —  i/y  =  4—1 

3.  Required  the  square  root  of  5ma  —  c+4ml/ma  —  c. 
We  have  a  =  5m9—  c,  and  ft  =  16m4—  16m*c.     Whence 

5m»_c+l/(5m»_c)»_(16m4—  IGm'c) 


=  4m  ; 


5m9 


and  we  have 


=  2m+l   ma — c, 


208  RADICAL  QUANTITIES. 

4.  What  is  the  square  root  of  ll-J-Gj/2?          Ans.  3-f-|/2. 

5.  What  is  the  square  root  of  7— 4T/3  ?  Ans.  2— j/3. 

6.  What  is  the  square  root  of  7— 2^/10  ?       Ans.  j/5 — y/2. 

7.  What  is  the  square  root  of  94-1-42^/5  ?       J.HS.  7-j-3j/5. 

8.  What  is  the  square  root  of  28-f  10|/3  ?         Jlns.  5-f-j/3. 

9.  What  is  the  square  root  of  rcp-f-2ma — 2mV/np-f-ma  ? 

J.TIS.  V«p-f-?7ia — »i. 

10.  What  is  the  square  root  of  ftc+^V'lo— 4»  ? 


11.  What  is  the  square  root  o 

.  5-f  31/^27 


12.  Find  the  value  of  \16-f301/— 1+\16— 301/— 1. 


Ans.  10. 


13.  Find  the  value  of  \  11-f  6^2-f  \7— 2^10. 

Ans.  3-f  !/5. 


14.  Find  the  value  of  X31-J-121/— 5-f- V-  1+ 4T/—5. 

4ns.  8+21/^5. 

15.  Find  the  value  of  Vn_|-l2|/2.  ^;w.  l-f-|/2. 


RATIONALIZATION. 

27G.  It  is  sometimes  useful  to  transform  a  fraction  whose  de- 
noaiinator  is  a  surd,  in  such  a  manner  that  tue  dp-nominator  shall 
become  rational.  The  fraction  is  thus  simplified,  because,  in  gen- 
eral, its  numerical  value  can  be  more  readiJy  calculated.  This 
transformation  is  usually  effected  by  multiplying  both  terpw  of  the 
fraction  by  the  same  factor. 


KATIONALIZATIOtf.  209 

277.  The  process  of  clearing  a  quantity  of  radical  signs  by 
multiplication,  is  called  Rationalization,  and  we  will  now  consider 
how  we  may  find  the  factor  which  will  rationalize  a  surd  quantity, 
in  some  of  the  more  important  cases. 

278.  To  find  a  factor  which  will  rationalize  any  mo- 
nomial. 

It  is  evident  that  the  factor  in  this  case  will  be  the  monomial  it- 
self, with  an  index  equal  to  the  difference  between  unity  and  the 
given  index.  For,  we  have  in  general  terms, 


1.  Rationalize  j/a. 

The  factor  is  |/a;  for,         |/aXl/«  =  «> 
a  rational  quantity. 

2.  Rationalize  x*. 

The  factor  is  x°  ;  for,   x°X^  =  &*  =  *• 

279.  To  find  a  factor  which  will  rationalize  a  bino- 
mial in  the  form  of  a±V6,  or  Va±V&,  m  and  n  being 
each  some  power  of  2. 

The  product  of  the  sum  and  difference  of  two  quantities  is  equal 
to  the  difference  of  their  squares.  Hence,  if  we  multiply  the  given 
binomial  in  this  case,  by  the  same  terms  connected  by  the  opposite 
sign,  the  indices  of  the  product  must  be  respectively  the  halves  of 
the  given  indices  ;  and  a  repetition  of  the  process  will  ultimately  ra- 
tionalize the  quantity,  provided  m  and  n  arc  any  powers  of  2. 

1.  Rationalize  a-j-|/c- 

The  factor  is  a  —  ^/c  ;  for  we  have 


2.  Rationalize  j/a  —  \/x. 

d/a—  4vAr)  d/a-f-V^)  =  a—  -/a;  ; 

(a—  -j/x)  (a-f  !/*)  =  a'—  x, 
a  rational  result;  and  the  complete  multiplier  is 


18* 


210  RADICAL   QUANTITIES. 

A  trinomial  may  be  treated  in  a  similar  manner,  when  it  con- 
tains only  radicals  of  the  2d  degree. 
3.  Rationalize 


T/5+J/2  -v/3 
•1/5+1/2  -f  y/3 

J/15 

1/10+      2—  j/6 


4— 2V/10 

16— 40=— .24 
a  rational  result ;  and  the  complete  multiplier  is 

(1/5+1/2-|-1/3)  (4— 21/10). 

Thus  we  perceive  that  it  is  necessary  simply  to  change  the  sign 
of  one  of  the  terms  of  the  trinomial,  and  multiply  by  the  result, 
repeating  the  process  with  the  product. 

SSO.  To  find  a  factor  which  will  rationalize  any 
binomial  surd  whatever. 

Let  the  binomial  be  represented  by  the  general  form, 

a7  ±  I*. 

Put  x  =  a  rand  y  =  b* ;  and  let  n  be  the  least  common  multiple 
of  r  and  s.  Then  x*  ±  y*  will  be  rational.  But  x-\-y  will  exactly 
divide  x*-{-yn  when  n  is  odd,  and  xn — yn  when  n  is  even ;  and  x — y 
will  exactly  divide  xn — yn  whether  n  is  odd  or  even;  (89).  These 
quotients  will  therefore  be  the  factors  that  will  rationalize  the 
respective  divisors.  Hence,  let  q  represent  the  required  factor; 
then 


CO  q  =  -jry-'  for  a r  -j-6 '",  when  n  is  odd ; 

x*—i/*  ±-       L. 

Ov  <?  =  — ^- — »  for  ar  -|-6*,  when  n  is  even ; 

x* — y*        —    - 

(°)  q  = >  for  a  r  — Z> ' ,  »i  being  odd  or  even. 


RATIONALIZATION.  211 

1.  Rationalize  the  binomial  a*-f-6*. 

Since  n  =  6,  an  even  number,  we  have  from  (2), 

9  =          ~=  ~jj       | 
the  factor  required ;  and 

the  rational  result. 

The  foregoing  methods  may  be  applied  in  the  solution  of  the  fol- 
lowing 

EXAMPLES. 

1.  Reduce  —  to  a  fraction  whose  denominator  shall  be  rational. 

Ans. 


-— 
2.  Reduce  -  to  a  fraction  whoso  denominator  shall 


be  rational.  21/15—31/1  0 

An..  -  g  -  • 

3.  Reduce  —  to  a  fraction  whose  denominator  shall  be  rational. 


1/2 
4.  Reduce  T/Q  to  a  fraction  whose  denominator  shall  be  rational. 


V72 


Ans. 

3 

5.  Reduce  -    to  a  fraction  whose  denominator  shall    be 

1/7+1/3 

rational.  Am    5d/7—  1/3) 

4 

6.  Reduce  —  }—L  —  to  a  fraction  whose  denominator  shall  be 

I/  a  —  i/c 

rational.  a-f-l/oc 


212  KADICAL   QUANTITIES. 

7.  Reduce  !_  —  JLk__  to  a  rational  denominator. 
1/11—  v/5 


Q 

8.  Reduce    /i]i     /"g  to  its  simplest  form.  .4ns.  l/ll  — 


9.  Reduce     ,  —  --  -^  —  to  its  simplest  form.    Ans.  4-f-  ^  15. 
10.  Reduce  --  -,  -  to  its  simplest  form.       "S*   -  - 


11.  Simplify  (8+1/8)  (8+T/5XT/5-2) 
(5-1/5)  (v/8+1) 


12.  Simplify  _ 

1-fa—  1/1—  a'  a 

13.  Find  the  factor  which  will  rationalize  -j/5  —  V2. 

Ans. 


, 

14.  Reduce  —  .  -  _  to  a  rational  denominator. 


Ans    g'^+aa 


RADICAL  EQUATIONS. 

281.  A  Radical  Equation  is  one  in  which  the  unknown  quan- 
tity is  affected  by  the  radical  sign. 

282.  In  order  to  solve  a  radical  equation,  it  is  necessary  in  the 
first  place  to  rationalize  the  terms  containing  the  unknown  quantity. 
In  case  of  fractional  terms,  this  may  be  effected  in  part  by  methods 
already  explained.     But  the  process  is  commonly  one  of  involution. 

The  following  are  examples  of  simple  equations  containing  radical 
quantities. 


RADICAL    EQUATIONS.  213 

1.  Given  1/x-j-ll-fl/o;  —  4  =  5  to  find  x. 

OPERATION. 

y  ^+114-1/^=4  =  5  ; 


by  transposition,  1/x-j-ll  =  5— V x— 4  ; 

squaring  and  reducing,  x-|-ll  =  x-f  21 — lOl/x — A ; 

transposing  and  reducing,  l/x — 4    =  1 ; 
squaring  and  reducing,  x  =  5. 


T-  — 

2.   Given  r  -  -  =  -  r  —  to  find  x. 


OPERATION. 


—x  —  5  __ 


Multiplying  both  terms  of     2x—  5—  2l/?=5]c    '  4cr—  35 


the  fraction  by  the  nume-  >  —  -  -  -  =  -  -  -  > 
rator,  (279),  J  '    _ 

clearing  of  fractions,  etc.,  2Vx*—5x  =  2x—  30  ; 

dividing  by  2,  l/x«—  5x  =  x—  15  ; 

squaring  and  reducing,  25x  =  225  ; 

whence,  a;  =  9. 

c  m-t/a  m 

3.  Given  -j—,  —  T  -\  --  -  —  =  -7  -  7  to  find  x. 
*    x—a         /x—/a 


The  least  common  multiple  of  the  denominators  is  x  —  a  = 
(^/x-t-i/a)  (i/x  —  i/a)  }  and  the  solution  will  be  as  in  the  fol- 
lowing 

OPERATION. 


p rr-> 

(c— m)1 


214  RADICAL    QUANTITIES. 

From  these  illustrations,  we  derive  the  following  precepts  for  the 
solution  of  radical  equations  : 

1. —  It  is  sometimes  advantageous  to  rationalize  the  denominate! 
of  a  fractional  term,  before  transposition  or  involution. 

2. — An  equation  should  be  simplified  as  much  as  possible  before 
involution  ;  and  care  should  be  taken  so  to  dispose  the  terms  in  the 
two  members  as  to  secure  the  simplest  results  after  involution. 

EXAMPLES   FOR  PRACTICE. 

Find  the  values  of  the  unknown  quantity  in  each  of  the  follow- 
ing equations  : 


/*=7. 

2.  x+3  =  1/V— 4x4-5~9*.  Ans.  x  =  5. 

3.  ^i,/— T-TO        ,_      *,„  ^ns   » =  16. 

a9— 4a 


4-  Ans-  x  = 


5.     _?L       I?  n-.  An*,  x  = 

\x 


,        - 

6.  —  —  —  -f  —-  -  =-—  -  .  Ans.  x  =  |. 
T/l-fa:        1/1—  x*       V\—  *' 

/—  —       -\/a-\-x*  —  8 

7.  T/c-f-arr:1-/^™ 

Vc+x 


8-    x—  ^9-f  xx*^S  =  3. 

9.    21/»—  21/x—  32  =  1/327  ^in*.  x  =  50. 

a  a-fc  c  /a-fc\* 

0-    7^=2-^=4  =  7^2* 


11. 


a3 — 3«*x  -j-  x*K  3a — a;  ~  a — x.         ^Lns.  x  =  3a — 1. 


EADICAL    EQUATIONS.  215 


c* 

13.    x4-  Vc*  —ax  =     .  Ans.  x  = 

l/ca— ax 


23-  •rVPST-  •      *"  -  -II 


15. 


17.    V  5+x+F  5— x  =  F  10.  Ans.  x  =  5. 

I~,  =      2a    -.  -4ns.  a  =  * 

1/i+i  3 


19.    x-\-a  =  ^  Itf+xV-V+aP.  Ans.  x  = 

4a 


—  2       41/6^—9 
20.  -/=  -  =  —  j=  --  Ans.  x  =  6. 

1/6x4-2      41/6x4-6 

4-4-  c 
21. 


/—  -  ^w*.  «  =  4. 


—  rrrr  -  .  -4n*.    x  =  -  . 

Vax+b       3Vax+5b 

4 


25.    —  —  —  —  --  -^  =  9.  ..  ~. 


97 

27.    J  -  =  -  .  AM.  x  =  a 


216  QUADRATIC   EQUATIONS. 


SECTION   V. 


QUADRATIC    EQUATIONS. 

983.  A  Quadratic  Equation  is  an  equation  of  the  second 
degree,  or  one  which  contains  the  second  power  of  the  unknown 
quantity,  and  no  higher  power  ;  as  3xa  =  48,  and  ax*  —  2bx  =  c. 
That  term  of  the  equation  which  does  not  contain  the  unknown 
quantity,  is  called  the  absolute  term. 

984.  Quadratic  equations  are  divided  into  two  classes  —  Pure 
Quadratics,  and  Affected  Quadratics. 


PURE   QUADRATICS. 

.  A  Pnre  Quadratic  Equation  is  one  which  contains  tho 
second  power  only,  of  the  unknown  quantity  ;  as  3.ra  —  7  =  20. 

NOTE.  —  A  pure  equation,  in  general,  is  one  which  contains  but  a  single 
power  of  the  unknown  quantity. 

986.  It  is  evident  that  by  the  rule  for  solving  simple  equations? 
every  pure  quadratic  may  be  reduced  to  the  form  of 

x*  =  a, 

in  which  a  may  be  any  quantity,  real  or  imaginary,  positive  or 
negative. 

Extracting  the  square  root  of  both  members  of  this  equation,  we 
have, 

x  =  -}-j/a  or  —  |/a 
Hence, 

Evdry  pure  quadratic  equation  has  two  roots,  equal  in  numerical 
value,  but  of  opposite  signs. 


x 


*  _  24 


1.  Given  —  -  ---  j  —  =  -  —  32,  to  find  the  values  of  x. 


" 


H 


PURE    QUADRATICS.  217 

I          -  ^ 

OPERATION. 

a»__4       ^3— 24      x9 

6     "        4       =2"  "     °2' 

clearing  of  fractions,     2x' — 8— 3x'-f72  =  6x*— 384 ; 
collecting  terms,      ~  "X2^  £  ^ '% .    7xa  =  448  j  3  ^ 

dividing  by  7,  xa  =  64 ; 

by  evolution,  x  —  ±8,  -4ns. 

We  have  therefore,   for  the  solution  of    pure   quadratics,   the 

following 

_^A  <~~  «o  ~  >'  •  ••  "~  <Jf  $.  ! 

EULE.  Reduce  the  equation  to  the  form  o/x*  =  a,  and  then  take 

die  square  root  of  loth  members. 

J6  7^ 

EXAMPLES    FOR    PRACTICE. 

Find  the  values  of  x  in  each  of  the  following  equations : 

1.  3x9— 16  =  x*-\-2.  Ans.  x  =  ±3 

2.  2x' — 54  =  126 — 3x*.  Ans.  x  =  ±6. 

3.  7z'-|-8  =57  +  3x*  + 15.  Ans.  x  =  ±4. 

4.  5x*-{-2^  —  2cx*-4-^« 


**—2c 

5.  ox'-j-l  =  (a— x)  (a-f-x).  -4ns.  cc  =  HZ  I/a— 1. 

oj.f.4      or— 4       10 

6.  — ~  H — j  =  -^-.  .4ns.  x  =  ±8. 

x — 4   '  x-f-4         3 

*-f2      x— 2       13 

7.  -  -^  +  —r^  =  -7T"  ^*.  «=  ±  10. 


x-4-a      x  —  a  3a 


10.        +      =+. 
a        x       c        x 


218  QUADRATIC    EQUATIONS. 

-ra      9-./Q         Ta      3 

12.  :  rfl£  _  ?L_i  =  L^g.     ^ 

13.  z9+2*  =  9+  —  . 


14.    —  —    -  =  1.  ^ns.  x  =  ±1.095445+. 


15.    ?=*  _ 


AFFECTED  QUADRATICS. 

287.  An  Affected  Quadratic  Equation  is  one  which  contains 
both  the  first  and  the  second  powers  of  the  unknown  quantity;    us 
2x*— 3.x  =  12. 

NOTES.  1.  The  two  classes  of  quadratics,  pure  and  affected,  are  some- 
times called,  respectively,  incomplete  and  complete  equations  of  the  second 
degree. 

2.  A  complete  equation,  in  general,  is  one  which  contains  every  power 
of  the  unknown  quantity,  from  the  first  to  the  highest  inclusive.  Thus 
a  complete  equation  of  the  third  degree  must  contain  the  first,  second, 
and  third  powers  of  the  unknown  quantity. 

288.  Every  affected  quadratic  equation  may  be  reduced  to  the 
general  form, 

x*+2ax  =  6, 
in  which  2a  and  b  are  positive  or  negative,  integral  or  fractional. 

For,  to  effect  this,  we  have  only  to  bring  all  the  terms  containing 
the  unknown  quantity  into  the  first  member,  and  all  the  known  terms 
into  the  second  member,  and  divide  the  result  by  the  coefficient  of  a2. 

289.  To  solve   a  quadratic,  suppose    it  first  reduced    to  the 
form, 

x*+2ax  =  b. 

To  both  members  add  a2,  or  the  square  of  one  half  the  coefficient 
of  x }  thus 

xa-f2a^-f  a*  =  a'-f  b. 


AFFECTED    QUADRATICS.  219 

The  first  member  is  now  a  complete  square.     Taking  the  square 
root  of  both  members,  we  have 

x+a  =  ±:l/a*-f&; 
whence  by  transposition, 

x  =  —  a±Va*+b. 

Thus  the  equation  has  two  roots,  which  are  unequal  in  all  cases 
except  when  a*-\-b  =  0  ;  in  this  case  we  shall  have 

x  =  a±:0, 

and  the  equation  is  said  to  have  two  equal  roots.    Thus  take  tho 
equation, 

*'—  lOz  =  —25. 
Adding  25  to  both  members, 


whence,  by  evolution, 

x—  5=±0; 
that  is,  x  =  5dbO  =  5  or  5. 

1.  Given  -  =  -j  —  -^—  =  -p-,   to  find  the  values  of  x. 
*"*" 

OPERATION. 


,  _ 

6 

clearing  of  fractions,  0x*-f  Gx'-f  12x-f  6  = 

reducing  terms,  x*-j-x  =  6  ; 

adding  (^)2  to  both  members,  x*+x-\-\  =  ~  • 

by  evolution,  ^+2  =  ±4  } 

whence,  x  =  2  or  —  3. 

Hence,  for  the  solution  of  an  affected  quadratic  equation  we  have 
the  following 

RULE.  I.  Reduce  the  given  equation  to  the  form  q/*xa-f-2ax  =  b. 
IT.   Add  to  loth  sides  of  the  equation  the  square  of  one  half  the  co- 
efficient o/"x,  and  thejlrst  member  will  be  a  complete  square. 

III.   Extract  the  square  root  of  both  members,  and  reduce  the 
resulting  equation. 


220  QUADRATIC   EQUATIONS. 

29O.  When  the  equation  has  been  reduced  to  the  form  of 

x*+  2ax  =  b, 
its  roots  may  be  obtained  directly  by  the  following  obvious  rule  : 

Write  one  half  the  coefficient  of  x  with  its  contrary  sign,  plus  or 
minus  the  square  root  of  the  second  member  increased  by  the  square 
of  one  half  this  coefficient. 

1.  Given  x*  —  Qx  =  55,  to  find  the  values  of  x. 

OPERATION. 


or,  x  =  3  ±8  =  11  or  —  5,  Ans. 


EXAMPLES  FOR   PRACTICE. 

Find  the  values  of  the  unknown  quantity  in  each  of  the  following 
equations : 

1.  x*+2x  =  15.  Ans.  x  =  3  or  —5. 

2.  x9 — 6x  =  16.  Ans.  x  =  8  or  —2. 

3.  x9— 20*  =  —96.  Ans.  x  =  12  or  8. 

4.  x9— 6x— 7  =  33.  Ans.  x  =  10  or  —4. 

5.  xf-28x-}-80  =  —115.  Ans.  x  =  15  or  13. 

6.  »»+6x+l  =  92.  Ans.  x  =  7  or  —13. 

7.  x*+I2x  =  589.  Ans.  x  =  19  or  —31. 

8.  x9— 6x+ 10  ==  65.  Ans.  x  =  11  or  —5. 

9.  aji+12a;+2  =  110.  Ans.  x  =  6  or  —18. 

10.  x9— 14x  =  51.  Ans.  x  =  17  or  —3. 

11.  x*+2%x-\- 19  =  0.  Ans.  x  =   -1  or  —19. 

12.  x*— &r+ 6  =  9.  Ans.  a;  =  8±21/8. 

13.  x*+8x  =  12.  Ans.  x  =  —4±%i/l. 

14.  aj«+12a;  =  10.  -4rw.  a  =  —  6  ±  V  46^ 

15.  3xa— 15x  =  —12.  Ans.  x  =  4  or  4-1. 

16.  4z«+ 12x  =  40.  Am.  x  =  2oT  —5. 
17    2af +28  =  18aj.  ^w.  *  =  7  or  2. 


AFFECTED    QUADRATICS.  221 

18.  (3^—5)  (2x— 2)  =  2(;ra4-]5).  Am.  x  =  5  or  —1. 

19.  (2z+ 2)(5x— 8)  =  (a;-|-l)(5x-f  4).  ^in«.  a  =  4  or  —1. 

20.  (3x+4)s  =  54x.  -4ns.  ac  =  f  or  f . 

21.  x*~  =  L  ^.  *  =  I  or  -TV 

lo       u 

22.  15*'  +  ^  =  5.  Ant.  x  =  f  or  — f . 


7        18* 

24 1 =  - .  Ans.  x  =  5  or  — 3. 

•    2(X— 1)  *  x1— 1      4 

4  5  12 

25.     — -^  H rr,  =  —TO-  Ans'  x  =  3  or  — f  • 

x-4-1  ^  x-J-2      x-j-3 


SECOND    METHOD    OF    COMPLETING   THE    SQUARE. 

.  It  frequently  happens  in  reducing  a  quadratic  to  the  form 
of  x*-\-2ax  =  Z>,  that  2a,  the  coefficient  of  x,  becomes  fractional, 
thus  rendering  the  solution  a  little  complicated.  In  such  cases  it 
will  be  sufficient  to  reduce  the  first  member  to  the  simplest  entire 
terms.  The  equation  will  then  be  in  the  form 

ax*+bx  =  c,  (1) 

in  which  a  and  b  are  integral  in  form,  and  prime  to  each  other,  and 
c  is  entire  or  fractional. 

To  render  the  first  term  of  equation  (1)  a  perfect  square,  multiply 
both  members  by  a;  thus, 

a*x*-\-abx  =  ac.  (2) 

Adding  -^  to  both  members,  we  have 

«V+ ate+l"  =  ac+?,  (3) 

*     \  &" 

where  the  first  member  is  a  complete  square.     Now  if  b  is  even,  — 

will  be  entire;  but  if  6  is  odd,  -  will  be  fractional,  a  result  which 
19* 


222  QUADRATIC    EQUATIONS. 

we  wisli  to  avoid.  To  modify  the  rule  to  suit  the  latter  case,  sup- 
pose equation  (3)  to  be  multiplied  by  4 ;  thus, 

4aV-f4a&x-f&a  =  4ac-f&'.  (4) 

The  first  member  is  now  a  complete  square,  and  its  terms  are  entire. 
Moreover,  we  observe  that  (4)  may  be  obtained  directly  from  (1)  by 
multiplying  (1)  by  4a,  and  adding  l>*  to  both  sides  of  the  result. 

Hence,  for  the  second  method  of  completing  the  square  in  the 
first  member,  we  have  the  following 

RULE.  I.  Reduce  the  equation  to  the  form  o/axa-|-bx  =  c,  where 
a  and  b  are  prime  to  each  other. 

II.  If  b  is  even,  multiply  the  equation  by  the  coefficient  of  x1, 
and  add  the  square  of  one  half  the  coefficient  of  x  to  Loth  members. 

III.  If  1)  is  odd,  multiply  the  equation  by  4  times  the  coefficient 
of  xa,    and  add  the  square  of  the  coefficient  of  x  to  loth  members. 

The  above  rule  may  be  considered  as  more  general  than  the  first ; 
for  if  applied  to  equations  in  the  form  of  xa-f-2ax  =  I,  the  opera- 
tion will  be  the  same  as  by  the  first  rule,  with  the  simple  modifica- 
tion of  avoiding  fractions  in  the  first  member,  when  2a  is  fractional. 

1.  Given  5xa — Qx  =  8,  to  find  the  values  of  x. 


OPERATION. 

5z'— Gx  =  8. 
Multiplying  by  5,  and  ad- ") 

ding  3a,  or  9,  to  both  mem-  [•         25z'T-30;c-}-9  =  49 ; 
bers,  3 

by  evolution,  5x — 3  =  ±7 ; 

whence,  5x  =  10  or  —4 ; 

or,  x  =  2    or  — J. 

2.  Given  ISa;' — 55x  =  350,  to  find  x. 

OPERATION. 

15x*_55z  =  350. 

Dividing  the  given  equation  by  5,  3x* — lice  =  70  ; 

multiplying  by  12,  and  ad-  j          36x,_132;c+12l  =  961 
ding  121  to  both  members,    ) 

by  evolution,  6x — 11  =  ±31 ; 

whence,  reducing,  x  =  7  or  — J^. 


AFFECTED    QUADRATICS.  223 

"We  will  now  apply  this  rule  to  an  equation  in  the  form  of 
x*-\-2ax  =  b,  where  2a  is  odd. 

3.  Given  x2 — 7x  =  44,  to  find  x. 

OPERATION. 

x'— Ix  =  44. 

Multiplying  by  4  times  1,  )  4x*_28x-|-49  =  225  • 
and  adding  T  to  both  sides,  ) 

by  evolution,  2x — 7  =  ±15  ; 

whence,  x  =  11  or  — 1. 

Thus  we  may  always  operate  in  such  a  manner  as  to  avoid  frac- 
tions in  the  first  member ;  and  indeed  in  the  second  member,  if  we 
first  reduce  both  members  of  the  equation  to  entire  terms. 


EXAMPLES   FOR  PRACTICE. 

Solve  each  of  the  following  equations  : 

1.  5x'-j-4x  =  204.  Ans.  x  =  6  or  —  J£. 

2.  5x'-f  4x  =  273.  Ans.  x  =  7  or  —  M. 

3.  7 x*— 20x  =  32.  Ans.  x  =  4  or  —  f . 

4.  6x'-f  15x  =  9.  Ans.  x  =  %  or— 3. 

5.  2xf— 5x  =  117.  Ans.  x  =  9  or  —  *£. 

6.  21x*— 292x  =  —500.  Ans.  x  =  11  j$  or  2. 

7.  6x'— 13x-f  6  =  0.  Ant.  x  =  f  or  j. 

8.  7x* — 3x  ;=  160.  Ans.  x  =  5  or  — 3^. 

9.  3x»— 53x  =  —34.  Ans.  x  =  17  or  f . 
10.  x»-{-13x— 140  =  0.  Ans.  x  =  7  or  —  20. 

12.  af+llx— 80  =  0.  Ans.  x  =  5  or  —16. 

13.  7x-f-     '"^     =  50.  ^ns.  a;  =  2  or  W- 

iU — ox 

14.  -       -  -j =.  =x  +  14.  Ans.  x  =  28  or  9. 


224  QUADRATIC   EQUATIONS. 

A/2rr  _  1  1  ^ 

15.  ;=26—  4s.  ^»«.  *=8  or 


5 


3x—  2       2x—  5       10 


TREATMENT  OP  SPECIAL  CASES. 

292.  Either  of  the  two  preceding  rules  is  sufficient  for  the 
solution  of  any  quadratic  equation  whatever.      There  are  certain 
cases,  however,  where  the  solution  may  be  much  simplified,  cither 
by  a  -modification  of  one  of  the  common  rules,  or  by  a  special  prep- 
aration of  the  example. 

293.  When  the  coefficient  of  the  highest  power  of 
the  unknown  quantity  is  a  perfect  square. 

In  this  case  the  equation  will  be  in  the  form  of 

aV+fcc  =  c.  (1) 

Let  the  quantity  to  be  added  to  complete  the  square  in  the  first 
member,  be  represented  by  t*.     Then 

oV+fcH-^  =  c+t\  (2) 

Now  in  any  binomial  square,  the  middle  term  is  twice  the  product 
of  the  square  roots  of  the  extremes.     Hence, 


•-=• 


And  equation  (2)  becomes  4 


which  may  be  used  as  the  formula  for  completing  the  square,  ir  this 
case.     Or  we  may  proceed  according  to  the  following 

RULE.  Divide  the  coefficient  of  x   by   twice  the  square  root  of  tA« 
coefficient  of  x2,  and  add  the  square  of  this  result  to  both  memo**"* 


AFFECTED    QUADRATICS.  225 

1.  Given  25x*— 20x  =  —8,  to  find  the  values  of  x. 

In  this  example  the  number  to  be  added  to  complete  the  square  is 


and  the  whole  operation  will  be 

25x*— 20x  =  —3, 
25x8— 20x-f4  =  l, 

5x— 2=±1, 

4x9       x       51 
2.  Given  —  —  —  =  ^-,  to  find  the  values  of  x. 

For  this  example  we  have 

MW)'- &)'=£' 

and  the  solution  is  as  follows : 

49  ~~2  =63' 

If!      f      49       100 
49^      Sf   '  64  ~   64  ' 
2x      7  10 

T'~~s~'±~st 

x  =  *f£  or  —  f  J,  ^ln». 


EXAMPLES  FOR  PRACTICE. 

1.  16z'  fl2x  =  10.  Ans.  x  =  ^  or 

2.  36x'—  5x  =  T3^.  Ans.  x=^  or  — 

3.  Six3—  12x  =  —  J.  Ans.  x  =  |  or 

49xa      6x       40 


,    841x9       58x 

5    -—  ---  -  =  11.  Ans.  x= 


6. 


=  2  or  - 


226  QUADRATIC    EQUATIONS. 

294.  When  the  equation  is  in  the  form  of 

x2+2ax=(2a+m)m.  (1) 

In  this  case  we  may  avoid  tedious  numerical  operations,  by  the 
use  of  the  auxiliary  quantity,  2a. 

1.  Given  x*  —  5x  =  6,  to  find  the  values  of  x. 

Put  2a  =  5  ;  then  2a+l  =  6; 
and  the  equation  becomes 

x*  —  2ax  =  2a-j-l  ;  whence, 

x3—  2ax-\-  a9  =  aa  -f-2a-f  1, 
x—  a=±(a+l), 
x  =  2a-|-l  or  —1, 
x  =  6  or  —  1,  Ans. 

2.  Given  xa-{-19x  =  92,  to  find  x. 

Assume  2a  =  19  ;  then  4(2a-|-4)  =  8a-f  16  =  92  ; 
and  the  solution  will  be  as  follows  : 

x3+2ax  =  8a-fl6, 
x'+2ax4-aa  =  a'+8a+16, 
x+a  =  ±(a+4), 

x  =  4  or  —23,  Ans. 

Let  it  be  observed  that  we  always  put  the  coefficient  of  x  equal 
to  2a.     Then  the  method  will  apply,  provided  the  second  member  is 


or  4a-}-4, 
or  6a+9, 
or  8a-f-16  ; 

or,  in   general,  2am-}-m9  ;   that  is,  any  multiple  of  2a   plus  the 
square  of  the  multiplier. 

EXAMPLES   FOE  PRACTICE. 

Solve  the  following  equations  : 

1.  x2  —  Ix  =  8.  Ans.  x  =  8  or  —  1  . 

2.  x*+llx  =  26.  Ans.  x  =  2  or  —13. 
•  3.  x8—  17»  =  60.  Ans.  x  =  20  or  —3. 

4.  .c'-f  21x  =  46.  ute.  x  =  2  or  —23. 


AFFECTED    QUADRATICS.  227 

5.  x9  —  75x  =  76.  Ans.  x  =  76  or  —  1. 

6.  x*+12x  =  385.  Ans.  x  =  5  or  —77. 

7.  x8—  325x  =  3350.  Ans.  x  =  335  or  —10. 
29o.  When  large  numerals  may  be  avoided  in  the 

operation,  by  the  use  of  an  auxiliary  quantity. 

As  all  the   examples   of  this  kind  can  not  be  included  in  any 
general  classification,  we  give  the  following  illustrations  : 

1.  Given  x«-f-9984x  =  160000,  to  find  the  values  of  x. 

Put  2a  =  10000  ;  then  2a  —16  =  9984,  and  32a  =  160000. 
Whence,  xf-f  (2a—  16)x  =  32a, 

x*_j.(2a—  16)*+(a—  8)3  =  aa+16a+64, 
x+(o-8)  =  ±(a+8), 

x  =  16  or  —10000,  Ant. 

2.  Given  x*-f  45z  =  9000,  to  find  x. 

Put  a  =  45  ;  then  200a  =  9000. 


4z'-f  4az-faa  =  aa+800a, 

2x+a  =  ±l/a(a+800)  =  1/45x845. 

Multiply  one  of  the  factors  under  the  radical  by  5,  and  divide  the 
other  by  5 ;  then  we  have 

2x+a  =  1/225x169, 
2x+l5  •  3  =  ±15  •  13, 

2x  =  15 -10  or— 15  •  16, 

x  =  75  or  —120,  Ans. 
3.  Given  16xf— 225*  =  225,  to  find  x. 

Put  15  =  a  ;  then  16  =  a-f-1 ; 
whence, 

4(a+l)V— 4«2O- 


2(a-f  1)«  =  2af-f  2a  or  —  2o, 

=  a(a-f-l)  or  —  a, 
x  =  15  or  —  j§, 


228  QUADRATIC   EQUATIONS. 


EQUATIONS  IN  THE  QUADRATIC  FORM. 

29G.  There  are  many  equations  which,  though  not  really  quad- 
ratic, or  of  the  second  degree,  may  be  solved  by  methods  similar  to 
those  employed  in  quadratics.  All  such  equations  are  reducible  to 
the  following  form  : 

x*n+2ax»  =  b  ; 

in  which  x  represents  a  simple  or  a  compound  quantity,  and  n  is 
positive  or  negative,  integral  or  fractional.  It  is  always  necessary 
that  the  greater  exponent  should  be  twice  tJie  less. 

1.  Given  x4—  16za  =  —28,  to  find  the  values  of  x. 

OPERATION. 
^__16a;a  =  —28. 

Adding  8a,  or  64,         x4—  16x'-|-64  =  36  ; 

extracting  the  square  root,        x9  —  8  =  ±6  ; 

by  transposition,  x9  =  14  or  2  ; 

whence,  x  =  ±1/14  or  zhj/2,  Arts. 

2.  Given  x  —  6x2  =  —  5,  to  find  the  values  of  x. 

OPERATION. 

x—  6z*  =  —5. 

Completing  the  square,  x  —  6a;*-j-9  =  4  ; 

by  evolution,  x~  —  3  =  ±2; 

or,  x3  =  5  or  1  ; 

involving  to  the  2d  power,  x  =  25  or  1,   -4ns. 

3.  Given  or-S-j-lOar-1  =  24,  to  find  the  values  of  x. 


OPERATION. 
arS-flOz-1  =24, 

Completing  square,          or2  -j-lOor  *  -j-25  =  49  ; 
extracting  square  root,  x~l-\-b  =  ±7; 

transposing,  x~l  =  2  or  —  12  ; 

taking  the  reciprocals,  x  =  ^  or  —  y^,  Am. 


AFFECTED    QUADRATICS.  229 

4.  Given  (x8  f  2x)a— 23(>3-f  2x)  =  —120,  to  find  the  values  of  x. 
This  equation  is  in  the  quadratic  form,  for  it  contains  the  first  and 
second  powers  of  the  compound  quantity,  xa-|-2x. 
For  convenience,  let  us  assume 

y  =  xa-f  2x. 
Then  by  substitution  in  the  given  equation,  wo  have 

if— 2Zy  =  — 120  j 
whence,  y«— 23y+AJA  =  y, 

y-V  =  ±1 
y  =  15  or  8. 

We  have  now  the  two  equations, 

x*+  2z  =  15  and  z'-f  2x  =  8, 
which  are  solved  as  follows  : 

x*+2x  =  15,  x*+2x  =  8, 

xa+2x-f-l  =  16,  ««+2aj+l  =  9, 

z+1  =  ±4,  as+l  =  ±3, 

x  =  3  or  — 5.  x  =  2  or  — 4. 

Thus  the  equation  has  four  roots, 

3,  -5,  2,  -4; 

ami  it  will  be  found  by  trial  that  any  one  of  these  four  values  will 
satisfy  the  given  equation. 

Equations  of  the  third  and  fourth  degrees  may  often  be  solved 
like  quadratics,  even  if  they  do  not,  at  first,  present  themselves  in 
the  quadratic  form,  like  the  last  equation.  If  any  equation  is  sus- 
ceptible of  such  a  solution,  it  will  be  found  to  contain  the  first  and 
second  powers  of  some  compound  quantity,  with  known  coefficients. 
To  determine  whether  this  be  the  fact  in  any  particular  case,  we 
may  proceed  as  follows  : 

Transpose  all  the  terms  to  the  first  member;  and  if  the  highest 
power  of  the  unknown  quantity  is  not  even,  multiply  the  equation 
through  by  the  unknown  letter,  to  render  it  even.  Then  extract 
the  square  root  to  two  or  more  terms,  as  the  case  may  require ;  and 
if  at  any  time  a  remainder  occurs,  which,  with  or  without  the  abso- 
lute term,  is  a  multiple,  or  an  aliquot  part  of  the  root  already  ob- 
tained, a  reduction  to  the  quadratic  form  may  be  effected.  Other- 
wise it  will  be  impossible. 
20 


230  QUADRATIC    EQUATIONS. 

5    Given  x4— 4x'— 14xa-f-36x-f  45  =  0,  to  find  x. 

SOLUTION. 

'—  14x'+36x-f-45  =  0|x*-2x 


2x»-2x          — 4x«-14xf 
— 4x'-f  4x* 


—  18xa-f36x-|-45 
Or  by  factoring,         _18(x8—  2x)-f  45 

Hence  the  given  equation  may  be  written  thus  : 
(x*—  2x)a—  18(xa—  2x)+45  =  0. 

Without  substituting    any  letter    for   the   compound   quantity, 
xf  —  2x,  the  remaining  part  of  the  operation  will  be  as  follows  : 

(x«—  2x)'—  18(xa—  2x)  =  —45, 
(a?—  2x)f—  18(x'-2x)+81  =  36, 
(x8—  2x)—  9  =  ±6, 

whence,  re*  —  2x  =  15,  (1) 

or,  x*—  2x  =  3.  (2) 

From  (1)  we  obtain  a:  =  5  or  —  3  )    * 

«     (2)  «       «  x  =  3  or  —  1  f 

6.  Given  x*-f-4ax>-f-2a*x  —  4a*  =  0,  to  find  x. 
As  the  highest  exponent  of  x  is  odd,  we  multiply  the  equation 
by  x,  and  obtain 

x4-f-4ax3-f  2aV-4a'a;  =  0. 
Taking  the  square  root  of  two  terms,  and  factoring  the  remainder, 

(x'+2axV—  2a*(x'-f2ax)  =  0. 
Assume  y=x*-f2ax  ;  then 

y*—2a*y  =  0, 
=  a4 


y  =  2a*  or  0. 
Hence  we  have  two  equations,  solved  as  follows  : 

x*+2ax  =  2aa,  xs-f-2ezx  =  0, 

x-j-o  ==  db«i/3,  xa=  —  2ax, 

x  =r  —  a±;a|/3.  x  =  —  2a. 

Hence,  x  =  —  a(l—  y/3),  —  «(l-{-1/3),  or  —  2a, 


AFFECTED    QUADRATICS.  231 

7.  Given    25x*—  6-f  ^  =  |,  to  find  x. 

The  two  extremes  in  the  first  member  are  perfect  squares.  We 
will  therefore  seek  for  a  middle  term  which  will  render  the  square 
complete.  This  will  be  twice  the  product  of  the  square  roots  of  the 
extremes}  or 


We  therefore  add  1  to  both  members,  and  solve  as  follows  : 


10zf—  1  = 


100*«:p30.r-ff  =  10-fj  =  y, 
lOxiF!  =  ±1 
10-c  =  5  or  —  2,  or  2  or 
*=  ±^or  :pi,   Ans. 

8.  Given  x-\-±yx=  21,  to  find  the  values  of  *. 
OPERATION. 


x+  tyx+4  =  25, 
.    -i/cc+2  =  ±5, 

l/x  =  B  or—  7, 
as  =  9  or  49,  Ans. 

It  should  be  observed  here  that  when  the  equation  contains  a 
radical,  as  in  the  last  example,  it  cannot  be  satisfied  by  the  roots 
obtained,  without  a  trial  of  signs.  The  roots  found  in  the  last  solu- 
tion are  9  and  49  j  but  we  have 

!/9  =  +3  or  —3, 
^49"=  +7  or  —7. 

Now  we  may  verify  the  given  equation,  if  we  take  j/ie  =-|-3  or 
—  7  j  but  not  otherwise. 


232  QUADRATIC   EQUATIONS. 

Thus,  putting  x  =  9  and  ^/x  =  3  -in  the  given  equation,  we  have 

9+12  =  21; 
also  with  x  =  49  and  i/x  =  — 7,  we  have 

49—28  =  21 ; 

and  the  equation  is  satisfied  in  both  cases. 
But  putting  x  =  9  and  -j/x  =  — 3,  we  have 

9—12  =  21; 
also  with  x  =  49  and  -j/x  =  -f-7,  we  have 

49+28=21; 
both  of  which  are  false. 

In  general,  it  will  be  found  that  a  radical  equation  can  be  satisfied 
by  each  of  the  roots  of  solution,  under  at  least  one  of  the  possible 

combinations  of  signs. 

2 
9.  Given  2i/x-f  — -  =  5,  to  find  x. 

l/x 

We  have  here  a  radical  equation  which  is  not  in  the  quadratic 
form.  In  such  cases,  it  is  generally  better  to  clear  the  equation  of 
radical  signs,  either  entirely  or  partially.  Thus, 


2x+2  = 
2x— 5!/x  =  —2, 
16aj—  40j/x+25  =  9, 

4V/X— 5  =  ±3, 
4j/x  =  8  or  2, 
l/x  =  2  or  j, 
x  =  4  or  |,  Ans 

• 

EXAMPLES  OF  EQUATIONS  SOLVED  LIKE  QUADRATICS. 

1.  x4— 34x3  =—  225.  Ans.  x  =  ±5  or  ±3. 

2.  x6— 35x3+216  =  0.  Ans.  x  =  2  or  3. 

3.  xfl— 4x8— 621  =  0.  Ans.  x  =  3  or  ^^23. 

4.  xl°+31x6— 32  =  0.  ^Tw.  x  =*2  or  1 . 


AFFECTED    QUADRATICS.  233 


5.  *«_  a    =  56.  Ans.  x  =  4  or 

6.  a*1—  23"  =  8.  ^n,.  *=V4  cr  VIT. 


?  L 

7.  2Qx»—  31x"  =  —12.  .4ns.  x  =  (f  )«  or  (f/. 

8.  3     »•—  lOrr  =  —3.  ^Lns.  x  =  27  or 


9.  X_|_5_if5  =  6.  Ans.  x  =  4  or  —  1. 

10.  O-f-12)i-Kse-{-12)l  ==  6.  -4ns.  x  =  4  or  69. 

11.  (z+a)^-f2i(:e-f  a)?  =  3Z»S.  -4ns.  x  =  i4—  a  or  81^4—  a. 

12.  x-f  l/5x+10  =  8.  .4ns.  x  =  18  or  3. 

13.  9x+4-f-21/9x-f4  =  15.  Ans.  x  =  |  or  |. 

14.  1/iO+a—  VicTfi  =  2.  Ans.    x  =  6  or  —9. 

15.  (x—  5)s—  3(>—  5)*  =  40.     Ans.  x  =  9  or  5-ff/(—  5)*. 

16.  2(1+  a—  x1)—  (1+oj—  a;1)^-}-^  =  0. 

Ans.  a?=|±|l/4r  or  J±JVTT 

17.  z_}_l6_31/a;-f-16  =  10.  ^Ins.  x  =  9  or  —12. 

18.  81*2+17+-\  =  99.  Ans.  x=  ±1  or  ±  J. 


19.  25*3+  6+-  =  -.  ^ns.  *  =  ±2  or 


20.  a:4+2x8—  7x9—  8a+12  =  0.  Ans.  x  =  1,  2,  —2,  or  —3. 

21.  x3—  8xa+19rr—  12  =  0.  ^ns.  x  =  1,  3,  or  4. 

22.  a4—  10x8-f  35x8—  50x-f24  =  0.    ^Ins.  x  =  1,  2,  3,  or  4. 

23.  x*—  8az8-f  8aV-f32a3a;—  9a4  =  0. 

^ns.  a;  =  a(2±;l/T3)  or  a(2±i/3). 

24.  y_2cy«-l-(c9-2y+2^=c». 

An*.  y  =     ± 


20* 


QUADRATIC    EQUATIONS. 
PROMISCUOUS   EXAMPLES   IN   QUADRATICS. 

1.  *•+!!*  =  80.  Ans.x  =  5  or  -16. 

2    3  3# — 3 3x — 6 


4          3  1  1 

2{V—  1)  —  4(>+l)  =  8*  ^ns-  *  =  3  or 


5  _ 

""  =  5  or  -J. 


"        —  An*.  x  =  I  or  —28. 


7.  a'- 

8.  ^-^ax+V  =  0.  .  x  = 


10  _i_8a!       20 

'  49  +  2l  ^  f  ^ns-  «  =  7  or  —  11  §. 

,,  ^a        12r, 

"  361~~l9"=  An*'  ^  =  152  or  76. 

19  8  16 

'  —    a  "      *"  *'                       Ans'  x  ~  3  or  1' 


13.  ^+li+l/?+li  =42.  ^MS.  cc  =  ±5  or  ±1/38. 

14.  ^—2^61/^=2^+6  =  11.    ^tn*.  x  =  1  or  1±  21/15. 

17#* 

15.  a:4+  —  -=  34x+16.         Ans.  x  =  2,  —2,  —8,  or  —£. 

16.  *-l=: 


. 

x 

x  =  4  or  7J. 


18 


.  x  Y 


3|/2- 


AFFECTED    QUADRATICS. 

~^~   _2+*' 
"2|/2~"V8   ' 


235 

(2±1/2)4. 


19    V*  — -— -Wl  — -=s 


,4ns.  x  = 


1— 


=1/?. 

l+l/l—.^        a1 


x  = 


22. 

23. 
24. 

25. 
26. 

27. 

28. 


2a 

Ofl  J 

" 


30 


81. 


32. 


=  x— 2. 


J.TW.  a;  =  5  or  3. 


— 9< 
=  756. 
— 13xw  =  —6. 


243  or 


=  c(l-*). 
=  352aV 


.  x  =  V|  or  Vf 
c1— 2 


'    (c+2)« 
or  sfcar — 7. 


a       6       a; 


a;  =  —  a  or  —  5. 
Ans.  x  =  ±o. 


x 


Ans.  x  = 


a-4-#         a — x 
a-fx-fl/^ax-f-x* 


.  x  =  it 


VX26— 6s 


236  QUADRATIC   EQUATIONS. 


SIMULTANEOUS  EQUATIONS  CONTAINING  QUADRATICS. 

3O7.  Having  treated  of  quadratics  containing  one  unknown 
quantity,  we  will  now  consider  certain  cases  of  simultaneous 
equations  where  one  or  more  is  of  a  degree  higher  than  the  first. 

2O8.  In  general,  the  solution  of  two  quadratic  equations,  involv- 
ing two  unknown  quantities,  depends  upon  the  solution  of  a  single 
equation  of  the  fourth  degree,  containing  one  unknown  quantity. 

To  show  this,  let  us  represent  the  two  equations  under  a  general 
form,  as  follows  : 

=  0,  (1) 

c'  =  0,  (2) 

in  which  the  coefficients  a,  b,  c,  etc.,  and  a',  I',  cr,  etc.,  may  have 
any  value,  positive  or  negative,  integral  or  fractional. 

Arranging  the  terms  in  these  equations  with  reference  to  cc,  and 
factoring,  we  have 

a*+(ay+c)x+by'+dy+c  =  0,  (3) 


Subtracting  (4)  from  (3),  we  have 

l(a-a')y+c-c'lx+(l-b'W+(d-d'^+(c--c>)  =-  0. 
By  transposition,  we  have 

[(a—  a'>+c—  c']  x  =  (6'—  6y-f(rf'— 
whence  we  obtain 

(l' 

This  value  of  x  substituted  in  either  equation,  (3)  or  (4),  will  give 
a  final  equation  involving  only  y.  Without  actually  making  this 
substitution,  which  would  lead  to  a  complicated  expression,  it  is 
obvious  that  the  resulting  equation  would  be  of  the  fourth  degree. 
For  the  value  of  x  is  in  the  form  of 


ri/+s 

in  which  y  is  involved  to  the  second  power.     Therefore  the  term 
containing  cca  in  (3),  or  (4),  must  involve  y  to  the  fourth  power. 


TWO    UNKNOWN   QUANTITIES.  237 

Hence,  two  equations,  essentially  quadratic,  and  containing  two 
unknown  quantities,  can  not,  in  general,  be  solved  by  the  rules  for 
quadratics. 

299.  There  are  certain  cases  where  simultaneous  equations, 
involving  one  or  more  of  the  unknown  quantities  to  a  higher  degree 
than  the  first,  may  be  solved  by  means  of  a  final  equation  in  the 
quadratic  form.  Most  of  the  examples  of  this  kind  are  embraced 
in  these  three  cases  : 

1st.  Where  one  of  the  equations  is  simple,  and  the  other 
quadratic. 

2d.  Where  both  of  the  equations  are  quadratic,  but  homogeneous. 

3d.  Where  one  or  both  of  the  equations  are  symmetrical, 
involving  the  different  letters  in  a  similar  manner  with  respect  to 
coefficients  and  exponents. 

The  following  are  illustrations  : 

1st.  SIMPLB  AND  QUADRATIC  EQUATIONS. 

The  solution  is  effected  in  this  case  by  the  ordinary  methods  of 
elimination. 


1.  Given  {  5*'~  Qxy  =  8  I  to  find  x  and  y. 
\  3x  —  2    =  6  J 


From  equation  (2),  x  = 

from(l), 


OPERATION. 

5*'—  Qxy  =  8,  (1) 

Sx  —  2y  =  6.  (3) 


72, 

•—  I2y  =  108  ; 


3  21 

4y-^  =  ±T; 

4y  =  12  or—  9; 

whence,  y  =  3  or  —  J. 

and  x  =  4  or 


238  QUADRATIC    EQUATIONS. 

2d.    HOMOGENEOUS  EQUATIONS. 

In  the  case  of  homogeneous  equations,  an  auxiliary  quantity  is 
employed  in  the  elimination. 

2.  Given  j  ^f^  IT  §  }  to  find  the  values  of  x  and  y' 


SOLUTION. 


Put  x  =  vy  }  then  the  given  equations  become 

- 

LI) 


Y-vy>  =  6,  or  y'  =  -JL  •  (1) 

- 


6  8 

whenec,  __  =  __, 

6+90  =  8e'—  4t>, 
St.*—  13»  =  G, 

«  =  2  or  -|. 
Taking  v  =  2,  equation  (2)  gives 

y  =  ±1  ;  whence  a;  =  ±2. 
Taking  v  =  —  |,  the  same  equation  gives 

8  3 

^  =  ±  —7^,  ',  whence  x  =  qi  —  .  % 

=  ^  l7^  1/7 

It  may  be  observed  here,  that  in  this  example,  as  in  all  other 
examples  of  simultaneous  equations,  the  different  sets  of  values 
which  are  capable  of  satisfying  the  equations,  will  be  found  by  tak- 
ing the  signs  in  their  order  ;  —  that  is,  the  upper  signs  should  be 
taken  throughout,  and  the  lower  signs  throughout.  Thus, 
when  y  =  +1,  x  =  -{-2  ; 


8  3 

7/  _     I  ~  _  _  . 

*'  "' 


TWO   UNKNOWN   QUANTITIES.  239 

3d.  SYMMETRICAL  EQUATIONS. 

When  the  equations  are  of  this  description,  they  may  be  solved 
by  taking  advantage  of  multiple  forms,  and  of  the  necessary  rela- 
tions existing  between  the  sum,  difference,  and  product  of  two 
quantities. 

3.  Given  \        ^  ~      ,  t  to  find  the  values  of  x  and  y. 
(       xy  =  21  ) 

OPERATION. 

x+y  =  10,  (1) 

xy  :=  21.  (2) 

Squaring  (1),  x*+2xy+i/*  =  100  ; 

subtracting  4  times  (2),        x*  —  1xy-\-y*  =  16  ; 
by  evolution,  x—  y  =  ±4; 

but  in  (1),  cc-fy  =  10; 

whence,  x  =  1  or  3  ; 

y  =  3  or  7. 


4.  Given  {  ^"^  =  ^l  to  find  the  values  of  x  andy. 
(  x*+xy+y*  =  133  J 


OPERATION. 

Put  x~{~y  =  s)  an(^  V^cry  =p. 

Then  the  given  equations  become 

«-{-p=:19,  (1) 

«'-  p«  =  133.  (2) 

Dividing  (2)  by  (1),  s_p  =  7;  (3) 

whence  from  (1)  and  (3),  s  =  13, 

and  p  =  6  ; 

that  is,  x+y  =  13, 

and  xy  =  36. 

Proceeding  now  as  in  the  third  example,  we  have 

=  169> 


x—y=  ±5, 
x  =  9  or  4, 
=  4  or  9 


240  QUADRATIC    EQUATIONS.  ' 

5    Given  •}  X4  '  ^2  =      3  v  to  find  x  and  y. 

OPERATION. 


Put  o= 

then  a*  =  P\  and  y?  =  Q*  . 

Substituting  these  values  in  the  given  equations, 


From  (1)  P'-f  2P<2+  £'  =  36  j  (3) 

taking  (2)  from  (3),  2PQ  =  16  ;  (4) 

taking  (4)  from-(2),  P*—  2P£-f  (?a  =  4  ; 
by  evolution,  P—  ^  =  ±2  ; 


P=4or2; 
Q  =  2  or  4. 
whence  we  have 

x$  =  4  or  2,  y*  =  2  or  4 ; 

x8  =  64  or  8,  y  =  32  or  1024. 

x  =  ±8  or  ±2j/2. 

In  this  example,  the  auxiliary  letters  were  used  to  avoid  frac- 
tional exponents  in  the  operation.  This  practice,  however,  is  not  a 
necessity,  but  only  a  convenience.  The  auxiliary  letters  should  be 
made  to  represent  the  loivest  powers  of  the  unknown  quantities. 

6.  Given  \  X'+XV  =  208    /    ^  find  x  &nd  y 
(y+x^s^  1053  ) 

OPERATION. 

Assume  x5  =  P,  y3  =  Q ; 

then,  x^  =  Pa,  ^  =  Q* ; 

Substituting  these  values  in  the  given  equations,  and  factoring, 
P«(P+Q)=    208  =  13-16,  (1) 

C"(e-fP)  =  1053  =  13-81.  (2) 


TWO    UNKNOWN    QUANTITIES.  2-11 

Dividing  (2)  by  (1),  7n=IJj->  (3) 

J-J- 

op  4  f) 

or,  C  =  'T'  and  />=~<r  (5) 

Substituting  these  values  in  (1)  and  (2),  we  have 


.  j 

FromCC),  7>.  =  a:.=:64; 

from(T),  e'=y=:729;  ~/ 

whence,  x  =  ±8, 

and  #==±27. 

If  we  take  tlic  minus  sign  in  the  second  member  of  equation  (•*), 
£-•  ,  we  shall  obtain  /-•/»?.£/ 

*=±Sl/=y;y  =  ±27T/:K'. 

VX/*/U    (   XJ.,,=     8) 

7.  Given   ]    ,      ,       -  r    \  to  find  the  values  of  x  and  y.  /-  ['  s.  / 
t  a;  -py  =  loL  )  . 

OPEnATIO,. 

- 


v  -+•  1  .  f  ••* 


=  152.  (2) 

Cubing  (1),        ,  x'-f  3*yf  3^/2-f^8  =  512  j  (3) 

taking  (2)  from  (3),  SxV+3-ry*  =  360,  (4) 

ay(*H-y)  =  120;  (5) 

dividing  (5)  by  (1),  xij  =  15.  (6) 

Whence,  by  combining  (1)  and  (6)  as  in  the  3d  example, 

x  =  5  or  3,  y  =  3  or  5.  >-  -  V  ^ 

3OO.  For  examples  of  more  than  two  unknown  quantities,  ru> 
additional  illustrations  arc  necessary.  The  few  cases  which  lead  to 
a  final  equation  in  the  quadratic  form  are  to  be  treated  by  the  same  y  - 
methods  that  apply  to  the  preceding.  And  skill  in  the  management 
of  this  whole  class  of  examples,  must  depend  less  upon  precept 
than  upon  practice.  . 

21  Q 


242  QUADRATIC    EQUATIOXS. 

SOS.  As  ixiliary  to  the  solution  of  certain  questions,  particu- 
larly in  geometrical  progression,  we  give  the  following 

PROBLEM. — Given  x-\-y  =  *  and  :ry  =  p,  to  find  the  values  of  x* 
"HiA  J'3-\-y*i  x'-\~#*>  and  x*-\-y"i  expressed  in  terms  of  s  aud/>. 

SOLUTION. 

x+y  =±  *,  (1) 

xy=p.  (2) 

Squaring  the  first,  x*-\-'±xy-\-y*  =.  sa ; 

.-toy          =2;,; 


1st  result,  x*+<f  =  s*—  'lp.  (A) 

Multiplying  (A)  by  (1), 


subtracting  //^  Q       xy(x+y)          = 

2d  result,  ^8-f-^s  = 

Agciiu,  squaring  (-4), 


subtractin 


8d  result,  cc4-|-y4  =  «4—  4**^+2j>*.       (C7) 

)  b 


Multiplying  (A)  by 


subtracting 
4th  result. 


The  following  example  will  illustrate  the  use  of  these  formulas. 

1  .  Given    \         '  ^  ~  (-to  find  the  values  of  x  and  ?/. 

U<_^-2417) 


In  this  example  we  have 

s  =  9,  «»  =  81,    s4  =  6561. 
Hence,  from  (C),  we  have 

6561  —  324p+2pf  =  2417  ; 

p'_162p  =  —2072, 
p9—  162p-|-6561  =  4489, 
p—  81  =  ±67, 
sry  =p  =  148  or  14. 

If  we  take  :ry  =  148,  the  values  of  x  and  y  will  be  imaginary, 
Taking  xy  =  14,  with  the  equation  ^-(-y  =  9,  we  readily  obtain 
x  =  7  or  2,  y  =  2  or  7. 


3  or  -21. 
2. 


TWO    UNKNOWN    QUANTITIES.  243 

EXAMPLES   OP   SIMULTANEOUS   EQUATIONS. 

Find  the  values  of   tho  unknown   quantities  in  the  following 
groups  of  equations  : 

!      (x-y     =15)  jx=18or    121, 

\x-2f  =    Of  Ans-    ty  = 

=  1207.  * 

=   22) 

=  35) 

=  25  f 

=  48) 
a    =    3) 

=  336  ) 

=40)  '(     =  —  72or—  8. 

7.     j-y+Jf-    =130) 
(5x-     =  7x    i 

A  = 

x-y)  =   ^  ' 

9. 


(  5x  -      =  25  '  (     =  10  or  45. 


f  Sx'+xy  =  336  )  A       (x=      28  or    12, 

'(     =  —  72 


=  ±  6. 
=161) 


10.       -3-2^-^  =  ll.  (x=    ±l/|f 

^         =2 


=50)  Ans     <x=±4l/2  or  ±14, 

2?/s  =  60  3  1  y  =  ±3v/2  or  q:  10. 

C  3^+2^  =    68  )  (  o:=±4  or 


=    68  )  (x=±i  or  ±^K3, 

=  160  J  ''  |  y  —  ±  5  or  q^t/3. 

-.    =  12)  P=±3or±76 

9  ,  Am~  ]  1 

(  *#— -#  =511  (  w  =  -+-1  or  H -• 

~l/t) 


244  QUADRATIC    EQUATIONS. 

.y=±5  or  ± 


14.  -  =  21 }  =  ±4  or 

5  =    0  ) 


=      8l' 


as  =  6  or  9,  or  —  9±1/5 
y  =  4  or  1,  or  —  3ipj/5. 


1/31 
17.         1i 


18.     j-9+/  =  89) 

(    x-ij  =    3)  (     =  5  or  —  8. 


=  13  1  An*     f^  = 

=    6)  t= 


y=2or—  3. 

22.     i^+/  =  (*+Jf)^l.  ^ 

I    ^+y  =  4 

f  3xy—  2ay  =  1     .  .       Ans.       *=  ±  ^2  or  ^  ^ 

or  ±      "-  6. 


.  ^  = 

1    *        =  12      f  = 


25.  ^  2402     •    Ans. 


. 
x+y=        8)  (y=l  or  7or4q=l/—  105. 

26.     1+^=12} 


27. 

=  16  ) 


TWO    UNKNOWN   QUA 


28.     - 


29.    {Xt~y~(^),  =  ^} 

(  £*-f-#a  =  a 


=6 


30. 

31. 

/  xy         =  2 

32.  r v  I 

\3cf+y        =333) 

(  X+V  vy  =  a 


Ans. 


33. 


x  =  4  or  (7^3)* 
y  =  9  or  324. 

af  &' 


=35 


35. 


36. 


a;  = 


Jlns. 


^ifW. 


*  =  27or    8, 
y  -    8  or  27. 

(  x  =  8  or  2, 


=16 


) 

) 


-dns. 


37.  r; 


38. 


(    $       ^  ) 


=  23 

=  6 
, 
" 


21* 


udns.    •! 

(y-8. 


',  8,  —1,  or  —216  ; 
27, —216,  or— 1. 


246 


QUADRATIC   EQUATIONS. 


39 


Ans. 


=  ±>  ±,  or     i 


40. 


42. 


43. 


\  ' 

(  x  =  2744  or  8, 
|  y  —  9604  or  4. 

=  1009  ) 
=  582193  \  ' 

(  x  =  81  or  16, 

An8'  )  y  ==  16  or  81. 
i 

'-8x       =64) 
,    ,  }• 

y-2x%5=    4J 

jxV-f^  =  30| 

u-  \  tr 


jlns. 


irt.+i 
.  ~<t+( 


45. 

46. 
47. 

48 


=  3093  ) 
x-y   =       3J- 


=    7 


_  =  2  or  — 5. 

I         J>_ 

X~  2  °'    26' 

1         15 

/  =  -Q   or  T^' 
o  lo 


Z   =  -j    Or     -r-r. 

4          44 


THEORY    OF    QUADRATICS.  247 


THEORY  OF  QUADRATICS. 

.  Having  treated  of  the  practical  methods  of  solving  quad- 
ratic equations,  we  will  now  proceed  to  consider  certain  general 
principles  relating  to  quadratics. 

3O3.  Let  us  resume  the  general  equation, 

x*+2ax  =  b.  (A) 

If  we  solve  this  equation,  and  represent  one  root  by  r  and  the 
other  by  r1  ',  we  shall  have 

+  6,  (1) 


=  —a—V~a*+b;  (2) 

By  adding  these  equations,  and  also  multiplying  them  together,  we 

obtain 

r+r'  =  —  2«,  (3) 

rr'  =  -b.  (4) 

That  is, 

1.  —  The  sum  of  the  two  roots  is  equal  to  the  coefficient  of  x  taken 
with  the  contrary  sign. 

2  —  The  product  of  the  two  roots  is  equal  to  the  absolute  term 
taken  with  the  co.itrary  sign. 

3O4.  From  equations  (3)  and  (4)  in  the  last  article,  we  have 

2a  =  —  (r-j-r'),  and  b  =  —rr'. 
Substituting  these  values  in  (J.),  and  transposing  the  absolute  term, 

we  have 

x*—  (r-f  r')x+rr'  =  0  j 

or  by  factoring, 

(x—r)(x—r')  =  0.  Hence, 

If  all  the  terms  of  a  quadratic  equation  be  transposed  to  the  first 
member,  the  result  will  consist  of  two  binomial  factors,  formed  by 
annexing  the  two  roots  with  their  opposite  signs  to  the  unknown 
quantity. 

3O«>.  A  Quadratic  Expression  is  one  which  contains  the  first 
and  second  powers  of  some  letter  or  quantity. 

By  the  principle  established  in  the  preceding  article,  any  quadratic 
expression  whatever  may  be  resolved  into  simple  factors. 


^ 

' 


248  QUADRATIC    EQUATIONS. 

1.  Let  it  be  required  to  resolve  the  expression,  :t*-f-12.r — 45,  in- 
to simple  factors. 

Assume  rra-f-  I2x — 45  =  0  , 

This  equation  readily  gives  c4  f 

x  =  3,  x  =  —15. 
Hence,  aa-{-12.r — 45  =  (x — 3)  (x-f-15),  Ans.  -j 

2.  Separate  5xa — 8x-{-3  into  simple  factors  s 
We  first  separate  the  factor  5 ;  thus, 

r* 

We  may  now  factor  the  quantity  within  the  parenthesis,  a*s  iu  tiie 
last  example;  thus,  i\ 


4_        1 
~5— -5'     *>  \  <Q 


x     .vt>      ^ 


.       3 
«  =  lor-.    ^j  -^,         v. 


0  .   r        , 


And  the  given  quantity  is  factored  as  follows  :   •    x     d 
Sx'-^+S  =  5(x-l)  (^-J),  ^«,.  •• 


O      Q 
OJI     ^| 


h 


ii 

EXAMPLES.  ^ 

1.  Resolve  xa-|-2a; — 120  into  simple  factors. 

-'Z-    -4iw.  (a:— 10)  (x+ 12.) 

2.  Resolve  x^—Qx+ll  into  simple  factors. 

V*  Jf    •*  — "  "7  «x    s 

-  ^~  /  <tr?s  Ans.  (x — 2)  (x — 7) 

3.  Resolve  x'-j-Sx-j-lo  into  simple  factors. 

Ans.  OH-3)  (x-f-5). 

4.  Resolve  xa — 35z-f  300  into  simple  factors. 

V  -  ^>-£*-  ^«'  "  ^4«s.  (x— 15)  (x— 20). 

5.  Rcsolvcx* — —  —  —  into  simple  factors. 

.-j,-+.rs  J-,,K  _j 

X»  »  -•  r      f  i- 


THEORY,  OT   QUADRATICS.  249 

V_JJ-4-l-^_.JL__Jt 
/    —  ~  ,30      •p   -**  ""  S~  X 

6.  Resolve  15x*-f-19x-{-6  into  simple  factors. 

Am.  15(aH-§)  (x+I). 

7.  Resolve  ex7  —  2ax-\-c*x  —  2ac3  into  simple  factors. 

/     *    2a  \ 
^4ns.  c(x  --  -J(x-fc3). 

3OO.  The  same  principle  also  enables  us  to  construct  an  equa- 
tion whose  roots  shall  be  any  given  quantities.  This  is  done  by 
multiplying  together  the  two  binomial  factors,  which,  according  to 
the  principle  in  question,  the  required  equation  must  contain. 

1.  Find  the  equation  whose  roots  shall  be  \  and  —  \. 

Factoia, 


or,  Gx'-f-z—  1  =  0,  Ans. 


EXAMPLES. 

1.  Find  the  equation  whoso  roots  shall  be  6  and  —  15. 

Ans.  x*Sx—  90  =  0. 


2.  Find  the  equation  whose  roots  shall  be  3  and  —  15. 

Ans.  xa--12x—  45  =  0. 


3.  Find  the  equation  whose  roots  shall  be  16  and  9. 

AM.  a;8—  25x-f  144  =  0. 

4.  Find  the  equation  whose  roots  shall  be  84  and  —  1. 

(V  -  Ans.  x9—  83x—  84  =  0. 

5.  Find  the  equation  whose  roots  shall  be  |  and  —  £. 

-       i 


6.  Find  the  equation  whose  roots  shall  be  |  and  —  ^. 

(Y-  ,    IT*    i 

An,  ,.____  =0. 

7.  Find  the  equation  whose  roots  shall  be  ^  and  \. 

Ans.  8x9—  G.r-j-1  =  0. 

8.  Find  the  equation  whose  roots  shall  be  2a  and  —  c. 

l^^^l      -       '  Ana.  x*—  tfa—c^xfiZac  ^=  0. 


250  QUADRATIC    EQUATIONS. 


DISCUSSION   OF   THE   FOUR   FORMS. 

3O7.  In  the  general  equation  -t2-f  2ax  =  b,  the  coefficient  of  x, 
as  well  as  the  absolute  term,  may  be  either  positive  or  negative. 
Hence,  to  represent  all  the  varieties,  with  respect  to  signs,  we  must 
employ  the  four  forms,  as  follows: 

x*+2ax  =  -f-fe,  (1) 

xa— 2ax  =  +6,  (2) 

x*+2ax  =  —b,  (3) 

x*—2ax  =  — -b.  (4) 

From  these  equations  we  obtain 

x  =  —a+i/'cf+b,  (1) 

x  =  -j-a±l/aa-h&,  (2) 

a;  =  —  a±l/a9^  (3) 

x  =  -fartl/a2— 6.  (4) 

"We  may  now  consider  what  conditions  will  render  these  roots  real 
or  imaginary,  positive  or  negative,  equal  or  unequal. 

SO8.  Heal  and  imaginary  roots. 

In  the  first  and  second  forms,  the  quantity  aa-f-6,  under  the  rad 
ical,  is  positive,  and  the  radical  quantity  is  therefore  real.  But  in 
the  third  and  fourth  forms,  the  quantity  a2 — b,  under  the  radical, 
will  be  negative  when  b  is  numerically  greater  than  a' ;  in  which 
case  the  radical  quantity  is  imaginary.  Hence, 

1. — In  ea.ch  of  the  first  and  second  forms,  both  roots  are  always 
real. 

2. — In  each  of  the  third  and  fourth  forms,  both  roots  are  imagi- 
nary when  the  absolute  term  is  numerically  greater  than  the  square 
of  one  half  the  coefficient  of  x  ;  otherwise  they  are  real. 

f;KO9.  Positive  and  negative  roots. 

Since  a~-\-b  >  a3    and   a2 — b  <  a2,  we  have 

vV-f-6  >  a  and  l/aa— b  <  a 

It  follows,  therefore,  that  the  signs  of  the  roots  in  the  first  and 
second  forms  will  correspond  to  the  signs  of  the  radical ;  but  the 


THEORY    OF    QUADRATICS.  1 

signs  of  the  roots  in  the  third  and  fourth  forms  will  correspond  to 
the  signs  of  the  rational  parts.  Hence, 

1. — In  each  of  the  first  and  second  forms,  one  root  is  positive  and 
the  other  negative. 

2. — In  the  third  form  both  roots  are  negative,  and  in  the  fourth 
form  loth  roots  are  positive. 

31O.  Equal  and  unequal  roots. 

It  is  obvious  that  in  the  first  and  second  forms  the  two  roots  are 
always  unequal  j  for  in  each  of  these  forms,  one  root  is  numerically 
the  sum  of  a  rational  and  a  radical  part,  and  the  other  the  difference 
of  the  same  parts. 

The  same  may  be  said  of  the  third  and  fourth  forms,  if  we  ex- 
cept the  case  where  a2  =  b  ;  in  which  case  the  roots  are  eqnal,  and 
we  have,  for  the  third  form, 

x  =  — a±0  =  — a  or  — a, 
and  for  the  fourth  form, 

x  =  -f-a±0  =  -fa  or  -\-a.  Hence, 

1. — In  each  of  the  first  and  second  forms,  the  two  roots  are  always 
unequal. 

2. — In  each  of  the  third  and  fourth  forms,  the  roots  will  be  equal 
tchen  the  absolute  term  is  numerically  equal  to  (he  square  of  one  half 
the  coefficient  of  x ;  otherwise  they  will  be  unequal. 

In  the  first  and  third  forms,  the  negative  root  consists  of  the  sum 
of  the  rational  and  radical  parts ;  while  in  the  second  and  fourth 
forms,  the  positive  root  consists  of  the  sum  of  the  two  parts.  Hence, 
if  we  exclude  the  case  of  equal  roots, 

3. — In  the  first  and  third  forms  the  negative  root  is  numerically 
greater  tlian  the  positive. 

4 — In  the  second  and  fourth  forms,  the  positive  root  is  numerical- 
ly greater  than  the  negative. 

The  principles  which  we  have  now  established,  respecting 
the  roots  of  quadratic  equations,  are  all  that  are  of  importance, 
either  theoretically  or  practically. 


252  QUADRATIC   EQUATIONS. 


DISCUSSION  OF  PROBLEMS. 

311.  In  the  solution  of  particular  problems  involving  quadrat- 
ics, we  shall  find  that  in  certain  cases  both  roots  of  the  equation 
will  answer  the  conditions  of  the  problem,  while  in  other  cases  only 
one  of  the  roots  is  admissible. 

The  reason  is,  that  the  algebraic  expression  is  more  general  in  its 
meaning  than  ordinary  language  ;  and  thus  the  equation  which  rep- 
resents the  conditions  of  the  given  problem,  will  sometimes  be  found 
to  represent  the  conditions  of  other  analogous  problems. 

1.  A  man  bought  a  horse  for  a  certain  price.  Now  if  he  sells 
him  for  $24,  he  will  lose  as  much  per  cent,  as  the  horse  cost  ;  re- 
quired the  price  of  the  horse. 


Let  x  denote  the  price.     Then  x  X  -777-  ?  orT7"7T>  will  be  the  loss,  if 

1UU        1UU 

he  sells  him  for  $2-4.     Hence,  we  have    • 


or,  a:a-100.r  =  —  2400; 

a'—  lOOjc-f  2500  =  100  j 
whence,  x  —  50  =  zblO; 

or,  x  =  GO  or  40. 

Both  values  of  x  fulfill  the  conditions.     For, 

60X-60  =  36  ;  and  60—36  =  24. 
40X-40  =  16  ;  and  40—16  =  24. 

2.  A  person  bought  a  number  of  sheep  for  $240  •  if  he  haa 
bought  8  more  for  the  same  sum,  each  sheep  would  have  cost  $1 
less.  How  many  sheep  did  he  purchase  ? 

240 
Let  x  =  the  number  of  sheep  purchased  ;  then  -  =  cost   of 

one.     Had  he  purchased  8  sheep  more,  the  cost  of  one  would  havo 

240  240  240 

been—.     Hence,  -—  1  =  _  ; 

reducing,  x*'+Sx  =  1920  ; 

a»+8;c+16=  193(5; 

or,  3+4=  ±44; 

whence.  x  =  40  or  ?—  48. 


DISCUSSION   OF   PROBLEMS.  253 

In  this  case  only  the  first  value  of  x  is  admissible.  The  negative 
result,  —  48,  is  numerically  the  answer  to  the  problem  which  would 
be  formed  by  substituting  in  the  above,  the  word  more  for  the  word 
less,  and  the  word  less  for  the  word  more. 

INTEKPRETATION   OF  IMAGINARY  'RESULTS. 

•Ilti.  We  have  seen  that  when  the  absolute  term  of  a  quadratic 
is  negative,  and  numerically  greater  than  the  square  of  one  half  the 
coefficient  of  the  second  power  of  the  unknown  quantity,  the  roots 
of  the  equation  will  be  imaginary.  Now  the  imaginary  roots  will 
always  satisfy  the  equation,  and  it  is  necessary  to  ascertain  what 
they  indicate  respecting  the  conditions  of  the  problem  which  the 
equation  represents. 

1.  Let  it  be  required  to  divide  20  into  two  such  parts,  that  their 
product  shall  be  140. 

Let  x  =  one  part  ;  then  20  —  x  =  the  other.     Hence. 

x(2Q—x)  =  140  ; 

or,  x9—  20x  =  —140, 

xa—  20*+100  =  —40, 

x—  10  = 


x  = 

The  result  is  imaginary  ;  how  shall  it  be  interpreted  ?  Recurring 
to  the  problem,  we  find  that  the  greatest  possible  product  that  can 
be  formed  by  multiplying  together  two  parts  of  20,  is  10x10  = 
100,  the  product  of  the  halves  of  20.  Thus  we  find  that  the  prob- 
lem is  impossible. 

2.  A  farmer  would  inclose  50  square  rods  in  rectangular  form,  by 
a  fence  whose  entire  length  shall  be  24  rods.  Required  the  length 
and  breadth  of  the  inclosurc. 

Let  x  =  the  length,  and  y  =  the  breadth  } 
then  x-\-y  —  12, 

and  xy  =  50  ; 

x*+2xy+y>  =  144, 
**—  2xy+y*  =  —56, 


x—  y  =  ± 

whence,  x  =  6±1/—14,  y  =  G^V^ 

22 


254  QUADRATIC    EQU>\TIONT3." 

Thus,  again,  the  results  are  imaginary.  The  problem,  however, 
is  impossible.  For,  if  any  given  area  is  to  be  inclosed  in  rectangular 
form,  the  perimeter  will  be  the  least  when  the  figure  is  a  square. 
But  the  square  root  of  50  exceeds  7  j  hence  the  field  will  have  a 
perimeter  of  more  than  28  rods,  and  can  not  be  inclosed  by  a  fence 
24  rods  long.  We  conclude,  therefore, 

That  imaginary  roots  indicate  impossible  conditions  in  the 
problem. 


PROBLEM   OF   THE   LIGHTS. 

313.  To  illustrate  more  fully  the  rules  of  algebraic  interpreta- 
tion, we  present  for  discussion  the  following  general 

PROBLEM. — Find  upon  the  line  which  joins  two  lights,  A  and  B, 
the  point  which  is  equally  illuminated  by  them ;  admitting  that  the 
intensity  of  a  light  at  any  given  distance,  is  equal  to  its  intensity  at 
the  distance  1/divided  by  the  square  of  the  given  distance. 

-^  jr A c       B <y 

Let  a  represent  the  intensity  of  the  light  A  at  the  distance  1, 
and  b  the  intensity  of  the  light  B  at  the  distance  1. 

Let  c  denote  the  distance  AB,  between  the  two  lights. 
Assume  A  as  the  origin  of  distances,  and  regard  all  distances 
measured  from  A  toward  the  right  as  positive. 

Finally,  let  C  denote  the  point  of  equal  illumination,  and  let  x 
represent  the  distance  of  this  point  from  A.  Then  c — x  must  be 
the  distance  of  the  same  point  from  B.  That  is, 

AC  =  x, 
BC  =  c—x. 
But  by  the  conditions  of  the  problem,  the  intensity  of  the  light 

A  at  the  distance  x  is  —5-,  and  the  intensity  of  the  light  B  at  the 

'X 

distance  c — x  is =.     But  these  intensities  are  equal,  because 

(c-a;)1 


PROBLEM    OF    THE    LIGHTS.  255 

C  represents  the  point  of  equal  illumination  ;  hence  we  have  the 
equation, 


(c-x?  _b  . 
x*        ~~  a  ' 


By  reducing  this  equation,  we  obtain  two  values  of  cc,  as  follows  : 

/^-ve/^-  x 

4  ±  x  ^ 

Since  the  two  values  of  x  are  real,  and  also  unequal,  we  conclude, 
That  there  are  two  points  of  equal  illumination  on  the  line  AB, 
or  on  this  line  produced. 

This  is  evidently  the  conclusion  to  which  we  ought  to  arrive  by 
an  algebraic  solution  of  the  problem,  in  order  to  satisfy  its  condi- 
tions in  a  general  manner.  For,  whatever  may  be  the  relative 
intensities  of  the  two  lights,  there  must  always  be  one  point  of 
equal  illumination  between  them.  And  if  the  lights  are  of  unequal 
intensities,  there  must  be  another  point  of  equal  illumination,  in  the 
prolongation  of  the  line,  on  the  side  of  the  lesser  light. 

TVe  will  now  discuss  the  values  of  x,  under  several  hypotheses. 

1st    Suppose  a>  b. 

In  this  case,  both  values  of  x  are  positive;  therefore,  both  points 
of  equal  illumination  are  situated  to  the  right  of  A. 

The  first  value  of  x  is  less  than  c  ;  because  -7-—  -  =•  is  less  than 


unity,  being  a  proper  fraction.     This  value  of  x  is  also  greater  than 
one  half  of  c  ;  for  we  have 

l/a=|/a;  (1) 

and  since  a  >  6,  -j/a-f|/6  <  2^/a.  (2) 

Dividing  (1)  by  (2),  we  have, 


256  QUADRATIC   EQUATIONS. 


and  consequently, 

1 

Hence,  the  first  point  of  equal  illumination  is  at  C,  between  A  and 
13,  but  nearer  B  than  A. 

The  second  value  of  x  is  greater  than  r  •  for,  —  —  —  -  is  greater 

I/a  —  yb 

than  unity,  being  an  improper  fraction.     Hence,  the  second  point  is 
at  C',  in  the  prolqngation  of  the  line  beyond  B. 

These  conclusions  arc  evidently  correct.  For,  the  supposition 
that  a  is  greater  than  b,  implies  that  B  is  the  feebler  light  ;  both 
points  should  therefore  be  nearer  B  than  A. 

£d.  Suppose  a  <  b. 

The  first  value  of  x  is  positive.  It  is,  moreover,  less  tlian  one 
lialf  of  c.  For  we  have 

l/a  =  v/a;  (1) 

and  since  a  <  5,  i/«-f  ;/&  >  2j/a.  (2) 

Dividing  (1)  by  (2),  we  obtain, 


and  therefore, 

Hence,  the  first  point  of  equal  illumination  falls  between  A  and  B, 
and  nearer  A  than  B,  as  it  should,  because  A  is  the  lesser  light. 

The  second  value  of  x  is  negative,  since  the  denominator,y/cr — j/Z>, 
is  negative. 

Now  in  the  statement  of  this  problem,  we  considered  distances 
reckoned  from  A  toward  the  right  as  positive  j  hencs,  according  to 
the  rule  for  interpreting  negative  results,  previously  established, 
(183),  we  must  consider  the  negative  result  in  this  case,  as  a 
distance  to  be  reckoned  from  A  toward  the  left.  Heiice,  the  second 
point  will  be  situated  to  the  left  of  A,  at  C".  And  this  is  as  it  should 
be,  because  A,  under  the  present  supposition,  is  the  lesser  light. 


PROBLEM    OF    THE    LIGHTS.  257 

3d.  Suppose  a  —  b. 

In   this   case,    the   first  value  of  x   is  positive,  and  equal   to  -^ 

u 

Hence  the  first  point  of  equal  illumination  is  midway  between  A 
andB. 

The  second  value  of  x  is  -~—  =  oo.      This  result  indicates  that 

there  is  no  other  point  of  equal  illumination  in  the  line  AB,  or  in 
AB  produced,  at  a  finite  distance  from  A. 

These  conclusions  are  obviously  correct.  For,  under  the  present 
supposition,  the  two  lights  are  equally  intense.  Hence  any  point, 
to  be  equally  illuminated  by  them,  must  be  equally  distant  from 
them ;  and  the  only  point  which  fulfills  this  condition  is  the  point 
midway  between  them. 

If,  however,  we  consider  a  and  b  as  two  varying  quantities,  at  first 
unequal,  but  continually  approaching  equality,  then  the  second  value 
of  x  will  become  greater  and  greater  by  degrees,  until  it  reaches 
infinity.  Under  these  conditions,  the  second  point  of  equal  illumi- 
nation will  continually  recede  from  A,  moving  toward  the  right  or 
toward  the  left,  according  as  a  is  greater,  or  less  than  b,  until  it  is 
finally  removed  to  an  infinite  distance.  In  this  view  of  the  case,  it 
is  sometimes  said  that  there  are  two  points  of  equal  illumination, 
under  the  hypothesis,  a  =.  b'}  one  point  being  at  an  infinite  distance 
from  A. 

4 tli.  Suppose  a  =  b  and  c  —  0. 

The  first  value  of  x  reduces  to =  0  :  hence  the  first  point 

Zj/a 

is  situated  at  A. 

The  second   value  of    x   is   - ,   the   symbol  of  indetermination ; 

(188,  4).  This  result  shows  that  there  arc  an  infinite  number  of 
other  points  equally  illuminated  by  the  two  lights. 

These  interpretations  are  evidently  correct.  For,  as  the  lights, 
under  the  present  hypothesis,  are  equally  intense,  and  both  situated 
at  A,  every  point  in  space  must  be  equally  illuminated  by  them 

5.  Suppose  c  =  0,  and  a  >  b  or  a  <  b. 
Both  values  of  x  now  reduce  to  0  j    and  the  common  rule  for 
interpreting  zero  might  lead  us  to  suppose  that  the  two  points  of 
22*  R 


258  QUADRATIC   EQUATIONS. 

equal  illumination  coincide  with  the  point  A.  But  this  conclusion 
is  not  strictly  correct;  for  it  is  obvious  that  when  two  lights,  of 
unequal  intensities,  occupy  the  same  place,  there  is  no  point  in  space 
equally  illuminated  by  them ;  not  even  the  point  in  which  they  are 
both  situated. 

Let  us  return  to  the  original  equation  (m),  which  truly  represents 
the  conditions  of  the  problem.  If  we  put  c  =  0,  the  result  is 

a  _  b 

xa  -  oT8'^  ^-2..C¥  +  >  - 

an  equation  which  can  not  be  satisfied  by  any  value  of  x  whatever, 
while  a  >  I  or  a  <  b.  For  by  substituting  any  value  for  x  we 
shall  always  obtain  two  unequal  fractions.  If  x  =  0,  the  two  mem- 
bers are  two  unequal  infinities. 

We  conclude,  therefore,  that  under  the  supposition,  c  =  0,  while 
a  and  b  are  unequal,  the  problem  fails  altogether,  and  is  impossible. 

Thus  we  learn  that  zero  may  be  the  answer  to  a  possible,  or  an 
impossible  problem.  And  whenever  we  obtain  this  symbol  as  the 
result  of  a  solution,  we  must  not  interpret  it  on  the  assumption  that 
the  thing  required  in  the  problem  is  possible ;  but  we  must  first 
determine  whether  the  conditions  are  rational  or  absurd,  by  consid- 
ering the  nature  of  the  problem,  or  by  substituting  zero  in  the 
original  equation. 


PROBLEMS   PRODUCING   QUADRATIC   EQUATIONS. 

.  It  will  be  found  that  some  of  the  following  problems 
may  be  solved  by  a  single  unknown  quantity,  while  others  require 
two.  Still  others  may  be  conveniently  solved  by  means  of  either 
one  or  two  letters.  It  is  left  to  the  judgment  and  skill  of  the 
learner  to  discover  the  mode  of  solution,  in  each  example,  which  is 
most  simple. 

1.  It  is  required  to  divide  the  number  14  into  two  such  parts, 
that  9  times  the  quotient  of  the  greater  divided  by  the  less,  may  be 
equal  to  16  times  the  quotient  of  the  less  divided  by  the  greater. 

Ans.  8  and  6. 


"-- 


PROBLEMS  PRODUCING  QUADRATICS.        259 

2.  A   company,  dining  at  an  inn,  agreed  to  pay  $3.50   for  the 

""entertainment;  but  before  the  bill  was  presented,  two  of  the  party 

j.  3  <-   left,  in  consequence  of  which  each  of  the  others  had  to  pay  20 

cents   more  than  if  all  had   been  present.      How   many   persons 

dined  ?  Ans.  7. 

.4' 

^  3.  There  is  a  certain  number,  which  being  subtracted  from  22, 
and  the  remainder  multiplied  by  the  number,  the  product  will  be 
117.  What  is  the  number  ?  An*.  13  or  9. 

4.  It  is  required  to  divide  the  number  18   into  two  such   parts, 
that  the  squares  of  these  parts  may  be  to  each  other  as  25  to  16. 

I   .// /  t~ ',  ••  '-o       Y  •  /  i  '  -  Ans.  10  and  8. 

5.  The  difference  of  two  numbers  is  4,  and  their  sum  multiplied 
by  the  difference  of  their  second  powers,  is  1600.     What  are  the 

f  ~  Q  (3  numbers  ?  2--  ,  -  '^)    -  /  £-  u-G  Ans.  12  and  8. 

?  6.  What  two  numbers  are  those  whose  difference  is  to  the  less  as 

4  to  3,  and  whose  product  multiplied  by  the  less  is  equal  to  504  ? 

Ans.   14  and  6. 

7.  A  man  purchased  a  field,  whose  length  was  to  its  breadth  as  8 
to  5.  The  number  of  dollars  paid  per  acre  was  equal  to  the  number 
of  rods  in  the  length  of  the  field ;  and  the  number  of  dollars  given 
for  the  whole  was  equal  to  13  times  the  number  of  rods  round  the 
field.  Required  the  length  and  breadth  of  the  field. 

Ans.  Length,  1 04  rods ;  breadth,  65  rods. 

7  y.    8.  There  is  a  stack  of  hay,  whose  length  is  to  its  breadth  as  5  to 

•  4,  and  whose  height  is  to  its  breadth  as  7  to  8.     It  is  worth  as  many 

cents  per  cubic  foot  as  it  is  feet  in  breadth ;  and  the  whole  is  worth 

at  that  rate  224  times  as  many  cents  as  there  are  square  feet  on  the 

'ritV     bottom.     Required  the  dimensions  of  the  stack. 

Ans.  Length,  20  feet ;  breadth,  16  feet ;  height,  14  feet. 
9.  There  is  a  number,  to  which  if   you  add  7  and  extract  the 
square  root  of  the  sum,  and  to  which  if  you  add  16  and  extract  the 
square  root  of  the  sum,  the  sum  of  the  two  roots  will  be  9.     What 
is  the  number?       >/L  -f-  t  l^?  ~tf  Ans.  9. 

NOTE. — Represent  the  number  by  ar— 7. 

y'  10.  A  and  B  together  carried  100  eggs  to  market,  and  each 
received  the  same  sum.  If  A  had  carried  as  many  as  B,  he  would 

"Y 


260  QUADRATIC    EQUATIONS. 

have  received  18  pence  for  them  ;    and  if  B  had  taken  as  many  as 
A,  he  would  have  received  8  pence.     How  many  had  each  ? 

Ans.  A  40,  and  B  60. 

11.  The  sum  of  two  numbers  is  6;  and  the  sum  of  their  cubes  is 
72.     What  are  the  numbers  ?  Ans.  4  and  2. 

12.  A  man  traveled  36  miles  in  a  certain  number  of  hours;  if  he 
had  traveled  one  mile  more  per  hour,  he  would  have  required  3 
hours  less  to  perform  his  journey.     How  many  miles  did  he  travel 
per  hour  ?    |£  *  -    ^  -t  3  )d  ^-t  Y-  ~  /  V        Ans.  3  miles. 

^  '       ;    13.  The   sum  of  two  numbers  is    100,  the  difference   of   their 
I/H  ~  'Square  roots  is  two  ;  what  are  the  numbers?        Ans.  36  and  64. 

14.  A  gentleman  bought  a  number  of  pieces  of  cloth  for  675 
dollars,  which  he  sold  again  at  48  dollars  a  piece,  and  gained  by  the 
bargain  as  much  as  one  piece  cost  him.  What  was  the  number  of 
pieces?  ^  V  *  £/rw  *  Ans.  15. 

r  ?  q  15.  A  merchant  sold  a  piece  of  cloth  for  39  dollars,  and  gained 
is  much  per  cent,  as  it  cost  him.     What  did  he  pay  for  it  ? 

Ans.  $30. 

16.  A  merchant  sent  for  a  piece  of  goods  and  paid  a  certain 
sum  for  it,  besides  4  per  cent,  for  carriage  ;  he  sold  it  for  $390,  and 
and  thus  gained  as  much  per  cent,  on  the  cost  and  carriage  as  the 
12th  part  of  the  purchase  money  amounted  to.  For  how  much  did 
lie  buy  it?  3$-  Ans.  8300. 


<017. 


^</£017.  From  two  towns,  396  miles  apart,  two  persons,  A  and  B,  set 
*out  at  the  same  time,  and  traveled  toward  each  other  ;  after  as  many 
days  as  are  equal  to  the  difference  of  miles  they  traveled  per  day,  they 
met,,  when  it  appeared  that  A  had,  traveled  216  miles.  How  many 
miles  did  each  travel  per  day  1  $  <  A>  Ans.  A,  36  ;  B,  30.  _ 

18.  Divide  the  number  60  into  two  such  parts  that  their  product 
shall  be  704.     ^  ,  ~  *  s  £-  7  0  V  Ans.  44  and  16. 

19.  A  vintner  sells  7  dozen  of  sherry  and  12  dozen  of  claret  for 
(     £50,  and  finds  that  he  has  sold   3   dozen   more  of  sherry  for  £10 

than  he  has  of  claret  for  £6.     Required  the  price  of  each. 

Ans.  Sherry,  £2  per  dozen:  claret,  £3. 

y  */  T.M  c 


PROBLEMS  PRODUCING  QUADRATICS.        261 

20.  A  set  out  from  C  towards  D,  and  traveled  7  miles  a  day. 

After  he  had  gone  32  miles,  B  set  out  from  D  towards  C,  and  went 

t+j  l-^vt/y^day  T'g  of  the  whole  journey;  and  after  he  had  traveled  as 

many  days  as  he  went  miles  in  a  day,  he  met  A.     Required  the' 

distance  from  C  to  D.  Ans,  76  or  152  miles. 


,    21.  A  farmer  received  $24  for  a  certain  quantity  of  wheat,  and 
an  equal  sum  at  a  price  25  cents  less  per  bushel  for  a  quantity  of 
'•'  barley,  which  exceeded  the  quantity  of  wheat  by  16  bushels.     How 
many  bushels  were  there  of  each  ? 

Ans.  32  bushels  of  wheat  and  48  of  barley. 

.22.  Two  travelers,  A  and  B,  set  out  to  meet  each  other,  A  leaving 

<J..   \*  ^  /  O- 

f)  at  the  same  time  that  B  left  D.     They  traveled  the  direct  road, 

.'  A-  jracfffif&YL8  miles  from  the  half-way  point  between  C  and  D;  and 

it  appeared  that  A  could  have  traveled  B's  distance  in  15|  days, 
and  B  could  have  traveled  A's  distance  in  28  days.     Required  the 
.    .         distance  between  C  and  D.  Ans.  252  miles. 

)d23f9'Tina  two  numbers,  whose  difference,  multiplied  by  the  differ- 
ence of  their  squares,  is  e>2,  and  whose  sum,  multiplied  by  the  sum 

of  their  squares,  is  272^$}  Ans.  5  and  3. 

'y7;/-  i?"^) 
24.  A  and  B  hired  a  pasture  at  a  certain  rate  per  week,  agreeing 

A.C*-  3that  each  should  pay  according  to  the  number  of  animals  he  should 
,  ,  ,  have  in  the  pasture.  At  first  A  put  in  4  horses,  and  B  as  many  as 

cost/  him  18  shillings  a  week ;  afterward  B  put  in  2  additional 
1  houses,  'and  found  that  he  must  pay  20  shillings  a  week.  At  what 

rate  was  the  pasture  hired  ?  Ans.  30  shillings  per  week. 

+  y  25.  If  a  certain  number  be  divided   by  the  product  of   its  two 

aigits,  the  quotient  will  be  2 ;  and  if   27  be  added  to  the  number, 
the  dibits  will  be  inverted.     What  is  the  number  ?  Ans.  36. 

26.  It  is  required  to  find  three  numbers,  such  that  the  difference 
of  the  fi~st  and  second  shall  exceed  the  difference  of  the  second  and 
third  by  6,   the  sum  of  the  numbers  shall  be  08,  and  the  sum  of 
the  squares  441.  Ans.  4,  13,  and  16. 

27.  What  two  numbers  are  those  whose  product  is  24,  and  whose 
sum  added  to  the  sum  of  their  squares  is  62  ?  Ans.  4  and  6. 


262  QUADRATIC    EQUATIONS. 

28.  It  is  required  to  find  two  numbers,  such  that  if  their  product 
be  added  to  their  sum,  the  result  shall  be  47 ;  and  if  their  sum  be 
taken  from  the  sum  of  their  squares,  the  remainder  shall  be  62. 

\.    ,    ^        -_   L  -j  i         t,        j  , 

7  /,  V  V    -^  ^*  -  fJ&  Ans.  7  and  5. 

••t      "*2r^_^~>      f      tn> 

7  NOTE. — In  many  examples  of  two  unknown  quantities,  giving  rise  to 
symmetrical  equations,  it  will  be  found  convenient  to  denote  one  of  the 
unknown  quantities  by  x+y,  and  the  other  by  x— y. 

29.  The  sum  of  two  numbers  is  27,  and  the  sum  of  their  cubes 
is  5103.     What  are  the  numbers  ?  An*.   12  and  15. 

30.  The  sum  of  two  numbers  is  9,  and  the  sum  of  their  fourth 
powers  is  2417.     What  are  the  numbers  ?  Am.  1  and  2. 

N  _  Cl.  The  product  of  two  numbers  multiplied  by  the  sum  of  their 
squares,  is  1248  ;  and  the  difference  of  their  squares  is  20.  What 
are  the  numbers  ?  Ans.  6  and  4. 


L    32.  Two  men  are  employed  to  do  a  piece  of  work,  which  they 
can  finish  in   12  days.     In  how  many  days  could  each  do  the  work 


I 


,  provide(i  it  would  take  one  10  days  longer  than  the  other 


Ans.  One  in  20  days  ;  the  other  in  30  days. 

-  The  joint  stock  of  two  partners  was  $1000  ;  A's  money  was 
in  trade  9  months,  and  B's  6  months  ;  when  they  shared  stock  and 
^'        gain/A'  receive^  $1,140  and  B  $640.     What  was  each  man's  stock  ? 

Ans.  A's,  $600  ;  B's,  $400. 

34.  A  speculator,  going  out  to  buy  cattle,  met  with  four  droves. 

In  the  second  were  4  more  than  4  times  the  square  root  of  one  half 

I  2the  number  in  the  first  ;  the  third  contained   three  times  as  many 

/as  the  first  and  second  ;  the  fourth  was  one  half  the  number  in  the 

third,  and  10  more;  and  the  whole  number  in  the  four  droves  was 

1121.     How  many  were  in  each  drove  ?      ^  ^"^  2.  y  i.  0 

Ans.  1st,  162;  2d,  40  ;  3d,  606  ;  4tli,  313. 

f  35.  Find  two  numbers,  such  that  if  the  sum  of  their  squares  be 
subtracted  from  three  times  their  product,  11  will  remain;  and  if 
the  difference  of  their  squares  be  subtracted  from  twice  their  prod- 
uct, the  remainder  will  be  14.  Aim.  3  and  5. 

36.  Divide  the  number  20  into  two  such  parts,  that  the  product 
of  their  squares  shall  be  9216.  Atm.   12  and  8. 


n 


y  *  96 


PROBLEMS    PRODUCING    QUADRATICS.  263 

37.  Divide  the  number  a  into  two  such  parts,  that  the  product 
"^       of  their  squares  shall  be  b. 

(  Greater  part,     |  +  -  (a9 
Ans.  1  .      i 

I  Less  part,          |  —  ^ 

*  ""  ^       38    The  greater  of  two  numbers  is   a9  times  the  less,  and  the 
/M  6 

product  of  the  two  is  i1.     Find  the  numbers.         Ans.  -  ,  and  ab. 

39.  A  certain  number  is  formed  by  the  product  of  three  consec- 
utive numbers;  and  if  it  be  divided  by  each  of  them  in  turn,  the 
sum  of  the  quotients  will  be  74.     What  is  the  number  ? 

.         j      120;  that  is,  4-5'6;  or 

(  —120  ;  that  is,  (—4)  •  (—5)  •  (—6). 

40.  An  engraving,  whose  length  was  twice  its  breadth  was  mounted 
frJt*  on  Bristol  board,  so  as  to  have  a  margin  3  inches  wide,  and  equal 

in  area  t'o  the   engraving,  lacking   36   inches.     Find   the  width  of 
t+JC  the  engiaving.  '^tf*  Ans.  12  inches. 

'41.  A  man  has  two  'square"  lots  of  unequal  dimensions,  containing 
',        together  25  A.  100  P.      If  the  lots  were  contiguous  to  each  other, 
it  would  require  28  J  rods  of  fence  to  embrace  them  in  a  single  in- 
7  ~      closure  of  six  sides.     Required  the  dimensions  of  the  two  lots. 

-^  Ans.  62  rods  and  16  rods,  50  rods  and  40  rods. 

-  '-  42.  A  person  has  £1300,  which  he  divides  into  two  portions,  and 
lends  at  different  rates  of  interest.  He  finds  that  the  incomes  from 
the  two  portions  are  equal  ;  but  if  the  first  portion  had  been  lent 
at  the  second  rate  of  interest  it  would  have  produced  £36,  and  if 
the  second  portion  had  been  lent  at  the  first  rate  of  interest  it  would 
have  produced  £49.  Find  the  rates  of  interest. 

A^s.  7  -and  6  per  cent. 
'  ^  se*s  out  fr°m   Condon  to  York,  and  B  at  the  same  time 


YL    2-/ 

from  York  to  London,  both  traveling  uniformly.     A  reaches  York 

25  hours,  and  B  reaches  London  36  hours,  after  they  have  met  on 
the  road.     Find  in  what  time  each  has  performed  the  journey. 

Ans.  A,  55  hours  ;  B,  66  hours. 

44.  A  owns  a  village  lot,  in  square  form,    containing  36  square 
rods  ;  B  owns  the  adjacent  lot  on  the  same  street,  which  is  also  a  M 


**- 


264  QUADRATIC    EQUATIONS. 

square,  but  greater  than  A's.  Now  if  A  should  purchase  all  the 
front  of  B's  lot,  so  as  to  extend  the  rear  boundary  line  of  his  own 
through  B's  lot,  parallel  to  the  street,  the  two  neighbors  would  pos- 
sess equal  quantities  of  land.  Find  the  length  of  one  side  of  B's 
lot.  Ans.  6(l-f-|/2)  rods. 

45.  There  are  three  numerical  quantities  having  the  following 
relations  to  each  other ; — the  sum  of  the  squares  of  the  first  and 
second,  added  to  the  first  and  second,  is  32  ;  the  sum  of  the  squares 
of  the  first  and  third,  added  to  the  first  and  third,  is  42 ;  and  the 
sum  of  the  squares  of  the  second  and  third,  added  to  the  second 

and  third,  is  50.     Required  the  quantities. 

f  1st,  3  or  — 4 ; 
Ans.    J  2d,  4  or  — 5  : 
(3d,5or-G. 

46.  "What  is'llic  side  of  that  cube  -which  contains  as  many  solid 
units  as  there  arc  linear  units  in  the  diagonal  through  its  opposite 
corners.  Ans.  V3. 

47.  It  is  required  to  find  two  quantities  such  that  their  sum,  their 
product,  and  the  sum  of  their  squares,  shall  all  be  equal  to  each 
other.  Ans.  £(3+1/^3),  and  i(3q:l/^). 

48.  Find  those  two  numbers  whose  sum,  product,  and  difference 
of  their  squares,  are  all  equal  to  each  other. 

Ans.  J(l3d:|/5)f  and  j(l^v^5). 

49.  Find  two  numbers,  such  that  their  product  shall  be  equal  to 
the  difference  of  their' squares,  and  the  sum  of  their  squares  shall 
be  equal  to  the  difference  of  their  cubes. 

Ans.  ±i|/5,  and  ^±1/5).    J 


PROPORTION.  265 


SECTION    VI. 


PROPORTION.  AND  THE  THEORY  OF  PERMUTATIONS 
AND  COMBINATIONS 

PROPORTION. 

31*>.  Two  quantities  of  the  same  kind  may  be  compared,  and 
their  numerical  relation  determined,  by  finding  how  many  times  one 
contains  the  other.  This  mode  of  comparison  gives  rise  to  ratio 
and  proportion. 

31G.  Ratio  is  the  quotient  of  one  quantity  divided  by  another 
of  the  same  kind  regarded  as  a  standard  of  comparison. 

There  are  two  methods  of  indicating  the  ratio  of  two  quantities. 

1st.  By  writing  the  divisor  before  the  dividend,  with  two  dots 
between  them;  thus, 

a  :  b 

indicates  the  ratio  of  a  to  b,  where  a  is  the  divisor  and  b  the 
dividend. 

2d.  In  the  form  of  a  fraction  ;  thus,  the  ratio  of  a  to  I  may  bo 
written  b 

a 

317.  A  Compound  Ratio  is  the  product  of  two  or  more  ratios. 
Thus, 

Simple  ratios,  I  '*  •  <J 

Compound  ratio,  ac  :  Id 

318.  The  Duplicate  Ratio  of  two  quantities  is  the  ratio  of. 
their  squares. 

319.  The  Triplicate  Ratio  of  two  quantities  is   the  ratio  of 
their  cubes. 

320.  Proportion  is  an  equality  of  ratios.     Thus,  if  two  quan- 
tities, a  and  b,  have  the  same  ratio  as  two  other  quantities,  c  and  dt 

23 


20'(J  PROPORTION. 

the  four  quantities,  a,  6,  c,  c7,  taken  in  their  order,  are  said  to  bo 
proportional. 

Proportion  may  be  written  in  two  ways ;  thus, 

a  1  b  I'.c  i  d, 
which  is  read,  a  is  to  b  as  c  is  to  d ;  or  thus, 

a  l  b  =  c  :  d, 

which  may  be  read  as  the  other,  or,  the  ratio  of  a  to  b  is  equal  to 
the  ratio  of  c  to  d.  The  second  method  of  writing  proportion  is  rec- 
ommended as  the  more  appropriate. 

.  A  Couplet  is  the  two  quantities  which  form  a  ratio. 

The  Terms  of  a  proportion  are  the  four  quantities  which 
are  compared. 

323.  The  Antecedents  in  a  proportion  are  the  first  terms  of 
the  two  couplets  j  or  the  first  and  third  terms  of  the  proportion. 

S^4r.  The  Consequents  in  a  proportion  are  the  second  terms  of 
the  two  couplets  ;  or  the  second  and  fourth  terms  of  the  proportion. 

32«>.  The  Extremes  in  a  proportion  are  the  first  and  fourth 
terms. 

326.  The  Means  in  a  proportion  are  the  second  and  third  terms. 

327.  When  the  first  of  a  series  of  quantities  has  the  same  ratio 
to  the  second,  as   the   second  has  to  the  third,  as  the   third  to  the 
fourth,  and  so  on,  the  several  quantities  are  said  to  be  in  continued 
proportion,  and  any  one  of   them  is  a  mean  proportional  between 
the  two  adjacent  ones.     Thus,  if 

a  :  I  —  b  :  c  =  c  :  d  =  d  :  e, 

then  tf,  b,  c,  d,  and  e  are  in  continued  proportion,  and  b  is  a  mean 
proportional  between  a  and  c,  c  a  mean  proportional  between  b  and 
d;  and  so  on. 

328.  One  quantity  is  said  to  vary  directly  as  another  when  the 
two  quantities,  by  reason  of  their  mutual   dependence,  have  always 
a  constant  ratio,  so  that  if  one  be  changed  the  other  will  be  changed 
in  the  same  proportion. 

Thus,  for  illustration,  suppose,  in  the  purchase  of  a  commodity,  a 
certain  quantity,  A,  costs  a  certain  sum,  B.  Now  if  the  price  of 
unity  remain  the  same,  it  is  evident  that  2  A  will  cost  2B ;  3 A  will 


267 

cost  3j8 ;  and  in  general,  mA  will  cost  mJS.  In  this  case  the  cost 
is  said  to  vary  directly  as  the  quantity. 

32O.  One  quantity  is  said  to  vary  inversely   as   another  when 
the  first  has  a  constant  ratio  to  the  reciprocal  of  the  other. 

330.  One  quantity  is  said  to  vary  as  two  others  jointly,  when  it 
has  a  constant  ratio  to  the  product  of  the  two. 

331.  The  Sign  of  Variation  is  the  symbol  oc  ;  thus,  the  expres- 
sion, A  oc  J3,  signifies  that  A  varies  as  B. 

From  the  definition  of  variation,  it  is  evident  that  the  expression, 
A  oc  B,  is  equivalent  to  the  proportion, 

A:£  =  m:l, 
where  m  is  a  constant.     This  proportion  gives 

A  =  mB. 

Hence  the  general  truth, 

If  A  vary  as  J5,  then  A  is  equal  to  B  multiplied  by  some  constant 
guantity. 

PROPOSITIONS   IN    PROPORTION. 

339.  A  Proposition  is  the  statement  of  a  truth  to  be  demon- 
strated, or  of  a  problem  to  be  solved. 

333.  A  Scholium  is  a  remark  showing  the  application  or  limit- 
ation of  a  preceding  proposition. 

334.  If  in  the  proportion 

a  :  b  =.  c  :  d, 
the  second  method  of  indicating  ratio  be  employed,  we  have 

-:-t 

which  is  the  fundamental  equation  of  proportion  ;  and  any  proposi- 
tion relating  to  proportion  will  be  proved,  when  shown  to  be  con- 
sistent with  this  equation. 

PROPOSITION    I. — In   every  proportion,   the  product  of  the  ex- 
tremes is  equal  to  the  product  of  the  means. 


268  PROPORTION. 

Let  a  :  b  ==  c  :  d  represent  any  proportion  j 

then  by  formula  (A\  -  =  - ; 

a       c 

clearing  of  fractions,  be  =  ad. 

That  is,  the  product  of  b  and  c,  the  means,  is  equal  to  the  prod- 
uct  of  a  and  d,  the  extremes. 

SCHOLIUM. — From  the  last  equation,  we  have 

The  first  mean, 

c  r 

The  second  mean, 


The  first  extreme, 

The  second  extreme, 

Hence, 

1st.  Either  mean  is  equal  to  the  product  of  the  extremes  divided 
by  the  other  mean.  (1) 

2d.  Either  extreme  is  equal  to  tJie  product  of  the  means  divided 
by  the  other  extreme.  (2) 

PROPOSITION  II. — Conversely : — If  the  product  of  two  quantities 
is  equal  to  the  product  of  two  others,  then  two  of  them  may  be  taken 
for  the  means,  and  the  other  two  for  the  extremes  of  a  proportion. 

Let  be  =   ad. 

Dividing  by  ac,  -  =  -  > 

a      c 

hence  by  formula  (A),  a  :  b  =  c  :  d, 

in  which  the  factors  of  the  first  product,  be,  arc  the  means,  and 

the  factors  of  the  second  product,  ad,  are  the  extremes. 

PROPOSITION  III. — If  four  quantities  be  in  proportion,  they  will 
be  in  proportion  by  ALTERNATION ;  that  is,  the  antecedents  will  be 
to  each  other  as  the  consequents. 

Let  a  :  b  =c:  «?; 

then  by  formula  (A\  -  =  -  J  (1) 

a      c 


PROPERTIES.  269 

be 
^multiplying  (1)  by  c,  —  =  d  ;  (2) 


dividing  (2)  by  bt  -  =     j  (3) 


hence,  a  :  c  =  b  :  d, 

in  which  a  and  c,  the  antecedents  of  the  given  proportion,  are  pro- 

portional to  b  and  d,  the  consequents  of  the  given  proportion. 

PROPOSITION  IV.  —  If  four  quantities  be  in  proportion,  they  wittbe 
in  proportion  by  INVERSION;  that  is,  the  second  will  be  to  the  fast, 
as  the  fourth  to  the  third. 

Let  a  :  b  =  c  :  d  ; 

b       d 

then  by  formula  (A),  —  =  —  > 

clearing  of  fractions,  bc  =  ad; 

hence  by  Prop.  II,  6  :  a  =  d  :  c. 

SCHOLIUM.  —  The  last  two  propositions  axe  but  modifications  of 
Prop.  II.  Thus  we  learn  that  from  every  equation  three  different 
forms  of  proportion  may  be  derived. 

Let          i  ad  =  be  ; 


then,         >  J       «        a  :  b  =  c  :  d} 

or,    »  *   a  :  c  =b  :  d;  ^ 

or,  b  :  a  =  d  :  c.  , 

PROPOSITION  V.  —  Quantities  which  are  proportional  to  the  same 
quantities  are  proportional  to  each  other. 

If  a  :  b  —  m  :  n,  (1) 

and  c  :  d  =  m  :  n,  (2) 

we  are  to  prove  that  a  :  b  =  c  id. 

From  (1),  -  =  -  ; 

a       m 

from  (2),  -  =  -; 

c        m 

b        d, 

hence,  —  =  —  ; 

a       c 

or,  a  :  b  =  c  :  d. 

23* 


270  PROPORTION. 

PROPOSITION  VI.  —  If  four  magnitudes  be  in  proportion,  they  must 
be  in  proportion  by  COMPOSITION  or  DIVISION;  that  is,  the  first  is  to 
the  sum  or  difference  of  the  first  and  second,  as  the  third  is  to  the 
iwm  or  difference  of  ike  third  and  fourth. 

If  a  :  b  =  c  :  d, 

we  are  to  prove  that       a  :  a+b  =  c  :  c+d. 


By  formula  U),  -  =  -:  (1) 


whence,  1-^ —  —  1-1 — ,  0J) 

a  c 

1-1=1-1;  (3) 

a  c 

from  (2),  ^ZL  _ 

a  c 

f  /ox  « ft          c ^ 

from  (o),  = j 

a  c 

hence  from  (4),  a  :  a-f-6  =  c  :  c-\-d; 

and  from  (5),  a  :  a — 6  =  c  :  c — d. 

SCHOLIUM. — In  4like  manner,  it  may  he  sho 


- 
£<+  ^PROPOSITION  VII.  —  If  four  quantities  be  in  proportion,  the  sum 

of  the  first  and  second  is  to  their  difference,  as  the  sum  of  the  third 
and  fourth  is  to  their  difference. 

If  a  :  b  =  c  :  d, 

we  are  to  prove  that   a-\-b  :  a  —  b  =  c-\-d  :  c  —  d. 
By  Prop.  VI,  a  :  a-\-b  =  c  :  c-fd;  (1) 

also,  a  :  a  —  6  =  c  :  c  —  d  ;  (2) 


from(1)>  _=. 

a  c 

a  —  b       c  —  d 

from  (2),  (4) 


C 


a  —  b       c  —  d 

dividing  (4)  by  (3),  =          ; 


whence,  a-j-6  :  a  —  b  =  c-\-d  :  c  —  d. 


PROPERTIES.  271 

PROPOSITION  VIII.  —  If  there  be  a  proportion,  consisting  of  three, 
or  more  equal  ratios,  then  either  antecedent  will  be  to  its  consequent, 
as  the  sum  of  all  the  antecedents  is  to  the  sum  of  all  the  consequents. 
Suppose          a  :  b  =  c  :  d  =  e  :  f=  g  :  h  =,  etc. 
Then  by  comparing  the  ratio,  a  :  b,  first  with  itself,  and  after- 
ward with  each  of  the  following  ratios  in  succession,  we  obtain 
ab  =  ba, 
ad  =  bcy 
af=be, 
ah  =-;  by,  etc.  ; 

whence,      a(b+d+f-\-h+  etc.)  =  b(a-\-c+e+g+  etc.), 
or,  a:b  =  a+c+e+g+  etc.  :  b+d-\-f+h+  etc. 


PROPOSITION  IX.  —  If  four  quantities  be  in  proportion,  the  term** 
of  eitlier  couplet  may  be  multiplied  or  divided  by  any  number,  and 
the  results  will  be  proportional. 

Let  a  :  b  =  c  :  d  ; 

b       d 
then,  -  =  -• 

And  since  the  value  of  a  fraction  is  not  changed  by  multiplying  or 
dividin"1  both  of  its  terms  by  the  same  number,  we  have 

=* 


•-: 

in  which  n  may  be  either  integral  or  fractional.     If  n  be  integral, 
we  have,  from  (1)  and  (2), 

na  :  nb  =  c  :  d,  (3) 

a  :  b  •=.  nc  :  nd  /  (4) 

in  which  the  terms  of  the  given  couplets  are  multiplied.    But  put 


n  =  —  ;  then  (3)  and  (4)  become 
m' 


"-    ;-=*.  .d,  (5) 

mm 

(6) 


m     m 
in  which  the  given  terms  are  divided. 


272  PROPORTION. 

PROPOSITION  X.  —  If  four  quantities  be  in  proportion,  either  the 
antecedents  or  the  consequents  may  be  multiplied  or  divided  by  any 
number,  and  the  results  iu  cccry  case  will  be  proportional* 

Let  a  :  b  =  c  :  d  ; 

then,  l-  =  *e.  (1) 

nit       nd 
whence,  —  =  —  ;  (2) 

y 


(3) 

in  which  to,  may  be  cither  integral  or  fractional.     If  n  be  integral, 

we  have  from  (2)  and  (3), 

a  :  nb  =  c  :  nd;  (4) 

na  :  b  =  nc  :  d}  0) 

in  which  the  given  antecedents  and  consequents  arc  multiplied.    Put 

n  =  —  ;  then  (4)  and  (o)  become 
m 

b_        d 

'  m  ~~      "  in 

a  c 

—  :  b  =  —  :  «  ; 
77i  m 

in  which  the  given  antecedents  and  consequents  are  divided. 

PROPOSITION  XI.  —  I/four  quantities  icliich  are  in  proportion,  be 
multiplied  or  divided,  term  by  term,  by  four  other  quantities  also  in 
proportion,  the  products,  or  quotients,  taken  in  order,  will  be  propor- 
tional. 

If  a:b  =  c:d,  (1) 

and  x  :  y  =  m  :  n,  (2) 

then  we  are  to  prove  that 

ax  :  by  t=  cm  :  dn, 

a      b        c     d 
and  —:—=  —  :  — 

x     y       m    n 

From  (1)  and  (2),  we  obtain 

ad  =  be  ',  (3) 

xn  =  ym  ;  (4) 


PROPERTIES.  273 

multiplying  (3)  by  (4),    (ax)(dn)  =  (%)(cm)  ;  (5) 


whence,  from  (5),  ax  :by  =.  cm  :  d/i  ; 

,  P        ,«,  a      &        c     cZ 

and  trom  (o)  _._.-._:_. 

x     y        m     n 

PROPOSITION  XII.  —  If  four   quantities  be  in  proportion,   Wee 
powers  or  roots  of  the  same  quantities  will  be  in  proportion. 
Let  a  :  b  =  c  :  d', 

b        d 

then  —  =  —  •  m 

a       c  W 

Raising  (1)  to  the  nth  power,  also  taking  tho  «th  root  of  the  same, 


i          i 

b»       d* 


(3) 


Hence  from  (2),  a*  :  b*  =  cn  :  d* ; 

JL      L         LI. 

and  from  (3),  a"  :  bn  =  cn  :  d* . 

PROPOSITION  XIII. — If  three  quantities  be  in  continued  propor- 
tion, the  product  of  the  extremes  is  equal  to  the  square  of  the  mean. 
Let  a  :  b  =  b  :  c ; 

then  by  Prop.  I,  ac  =  bb  =  b*. 

SCIIOLIUM. — Taking  the  square  root  of  the  last  equation,  we  have 
b  =  V  ac  ;  hence, 

The  mean  proportional  between  two  quantities  is  equal  to  the  square 
root  of  their  product. 

PROPOSITION  XIV. — If  three  quantities  be  in  continued  propor- 
tion, the  first  is  to  the  third,  as  the  square  of  the  first  is  to  the  square 
of  the  second  ;  that  is,  in  the  duplicate  ratio  of  the  first  and  second. 

Let  a  :  b  =  b  :  c  ; 

then  b9  =  ac ; 

multiplying  by  a,  ab*  =  a*c ; 

whence,  by  Prop.  II,         a  :  e  =  a9  :  6*. 

S 


•274  PROPORTION. 

PROPOSITION  XV. — If  four  quantities  be  in  continued  propor- 
tion, the  first  is  to  the  fourth,  as  the  cube  of  the  first  is  to  the  cube 
of  the  second;  that  is,  in  the  triplicate  ratio  of  the  first  and  second. 

Let  a  :  b  =  b  :  c •  =  c  :  d', 

then  ac  =  b9,  (1) 

and  c*  =  bd;  (2) 

multiplying  (1)  by  (2),  ac8  —  b*d ; 

whence,  by  Prop.  II,  a  :  d  =  b9  :  c* ; 

or,  a  :  d  =  a8  :  b*. 


PROBLEMS   IN   PROPORTION. 

To  show  some  of  the  applications  of  the  preceding  principles,  we 
give  the  following  problems  : 

1.  Find  two  numbers,  the  greater  of  which  shall  be  to  the  less 
as  their  sum  to  42,  and  as  their  difference  to  6. 

Let  x  =  the  greater,  and  y  =  the  less. 

By  the  conditions,    \  * 

C  x  :  y  =  x  — y  :  6.  (2) 

Prop.  V,                      x+y  :  42  =  x— y  :  6,  (3) 

Prop.  Ill,                x+y  :  x—y  =  42  :  6,  (4) 

Prop.  VII,                      2x  :  Zy  =  48  :  36,  (5) 

Prop.  IX,                            x:y=4:3,  (6) 

From  (1)  and  (6),  Prop.  V,  4  :  3  =  x+y  :  42,  (7) 

«     (2)   «    (6),     «      «  4:3  =  a:— ; y  :  6,  (8) 

"     (7),  Prop.  I,  x+y  =  56, 

"„   (8)     "        "  a:— y  =  8; 

whence,  x  = 


n.  «  Ans. 
and  y  =  24  j 

• 

2.  Divide  the  number  14  into  two  such  parts  that  the  quotient 
of  the  greater  divided  by  the  less,  shall  be  to  the  quotient  of  the 
less  divided  by  the  greater,  as  16  to  9. 


PROBLEMS.  275 

Let  x  =  the  greater,  then  14  —  x  =  the  less. 
x         14.  _  x 

By  the  conditions.  —  :  -  =  16  :  9. 

J  14  —  x         x 

Multiplying  terms,  Prop.  IX,  x*  :  (14—  z)a  =  16  :  9, 
extracting  square  root,  x  :  14  —  x  =  4  :  3, 

by  composition,  Prop.  VI,  x  :  14  =  4  :  7, 

dividing  consequents,  x  :  2  =.  4  :  1, 

x  =  S] 

whence,  a  c  Ans. 

14  —  x  =  6  ) 

3.  There  are  three  numbers  in  continued  proportion;  their  sum 
is  52,  and  the  sum  of  the  extremes  is  to  the  mean  as  10  to  3.      Re- 
quired the  numbers. 

Three  numbers  in  continued  proportion  may  be  represented  by 
x,  xy,  xy*  ;  for  we  observe  that  the  product  of  the  extremes  will 
then  be  equal  to  the  square  of  the  mean.  Hence, 

*  —  52  (1) 

=  10':  3.        (2) 

From  (2),  yf+l:y  =  10:3,         (3) 

or,  y-fl  :  2y  =  10  :  6,        (4) 

by  Prop.  VH,          #'H-2y+l  :  y'—  2^+1  =  16  :  4, 
taking  square  root,  #-|-l  :  y  —  1  =  4:2, 

by  Prop.  VII,  2y  :  2  =  6  :  2, 

or,  y:l  =  3:l, 

whence,  y  =  3  | 

and  from  (1),  ic  =  4  j 

4.  The  product  of  two  numbers  is  112  ;  and  the  difference  of 
their  cubes  is  to  the  cube  of  their  difference,  as  31  to  3.     What  are 
the  numbers  ? 

By  the  conditions,  \  ay  =  1  12, 

U»-  *  :   x-s  =  31  :  3.       2) 


t  x-L 
By  the  cond.Uons,  { 


»-  y*  :  (x-y)s  =  31  :  3.  (2) 

From  (2),  Prop.  IX,  x'-fry-f^9  :  tf—Zxy+y*  =  31  :  3,  (3) 

by  Prop.  VI,  3;ry  :  (x—  yj  =  28  :  3,  (4) 

by  substitution,  336  :f(x—t/y  =  28  :  3,  (5) 

whence,  /  ^  \x~yy  =  3^/  *  (6) 

or,  x—  y  =  6.  (7) 

From  (1)  and  (7),  we  obtain  x  =  14,  y  =  8. 


y  -«- 


276  PROPORTION. 

5.   What  two  numbers  are  those  whose  difference  is  to  their  sum 
as  2  to  9,  and  whose  sum  is  to  their  product  as  18  to  77  ? 
Let  x  and  y  represent  the  numbers. 

By  the  conditions,  \x~^  :  X+V  =    2:    °>  <1> 

(  x+y  :      xy  =  18  :  77.  (2) 

From  (1),  Prop.  VII,  2x  :  2y  —  11  :  7,  (3) 


From  (2),  by  substitution,     -y    :  --  =  18  :  77,  (5) 

by  Prop.  IX,  Ify  :  1  V  =  18  :  77,  (C) 

or,  18    :  lly  =  18  :  77,  (7) 

or,  1    :  y  =    1  :  7,  (3) 

whence,  y  —    7.        y.  »  // 

6.  Two  numbers  have  such  a  relation  to  each  other,  that  if  4  be 
added  to  each,  they  will  be  in  proportion  as  3  to  4  ;  i.nd  if  4  be 

/       subtracted  from  each,  they  will  be  to  each  other  as  1  to  4.     What 
are  the  numbers  ?  Ans.  5  and  8. 

7.  Divide  the  number  27  into  two  such  parts,  that  their  product 
shall  be  to  the  sum  of  their  squares  as  20  to  41. 

Ans.  12  and  15. 

8.  In  a  mixture  of  rum  and  brandy,   the  difference  between  the 
quantities  of  each  is  to  the  quantity   of   brandy,  as  100  is  to  the 
number  of  gallons  of  rum;  and  the  same  difference  is  to  the  quan- 
tity  of  rum,  as  4  to  the  number  of  gallons  of  brandy.      How  many 
gallons  are  there  of  each?         Ans.  25  of  rum,  and  5  of  brandy. 

9.  There  are  two  numbers  whose  product  is  320  ;  and  the  differ- 
ence of  their  cubes  is  to  the  cube  of   their  difference  as  61  to  1  . 
What  are  the  numbers?  Ans.  20  aj:d  16. 

NOTE.  —  In  the  last  example,  put  x+y  —  the  greater,  and  x—  y  =  the  less. 

10.  Divide  60  into  two  such  parts,  that  their  product  shall  be  to 
the  sum  of  their  squares  as  2  to  5.  Ans.  40  and  20. 

11.  There  are  two  numbers  which  are  to  each  other  as  3  to  2. 
,   v  If  6  be  added  to  the  greater  and  subtracted  from  the  less,  the  sum 


p 
/  f 


PROBLEMS.  277 

and  the  remainder  will  be  to  each  other  as  3  to'  1.     What  are  the 
numbers  ?  Ans.  24  and  16. 

12.  There  are  two  numbers  which  are  to  each  other  as  16  to  9, 
and  24  is  a  mean  proportional  between  them.  What  are  the  num- 
bers ?  4.  Ans.  82  and  18. 


13.  The  sum  of  two  numbers  is  to  their  difference  as  4  to  1,  and 
the  sum  of  their  squares  is  to  the  greater  as  102  to  5.  What  are 
the  numbers  ?  Ans.  15  and  9. 

\  ^^;;  0}4y  The  number  20  is  divided  into  two  parts,  which  are  to  each 
oilier  in  the  duplicate  ratio  of  3  to  1.      Find  the  mean  proportional 
-    L,     between  these  parts.  Ans.  6. 

15.  There  are  two  numbers  in  the  proportion  of  3  to  2  ;  and  if  6  be 
L  v      added  to  the  greater,  and  subtracted  from  the  less,   the  results  will 
be  as  9  to  4.     What  nrc  llio  numbers  ?  Ans.  39  and  26. 

^    16.  There   arc   three   numbers   in  continued  proportion.      The 
I  product  of  the  first  and  second  is  to  the  product  of  the  second  and 
third,  as  the  first  is  to  twice  the  second  ;    and  the  sum  of  the  first 
and  third  is  300.     What  are  the  numbers  ? 

Ans.  60,  120  and  240. 

17.  The  sum  of  the  cubes  of  two  numbers  is  to  the  difference  of 
their  cubes,  as  559  to  127  ;  and  the  square  of  the  first,  multiplied 
by  the  second,  is  equal  to  294.     What  arc  the  numbers  ? 

Ans.  7  and  6. 

18.  The  cube  of  the  first  of  two  numbers  is  to  the  square  of  the 
second  as  3  to  1,  and  the  cube  of  the  second  is  to  the  square  of  the 
first  as  96  to  1.     What  arc  the  numbers  ? 

Ans.  12  and  24. 

19.  Given  the  proportion  (a;-f-  1)4  :  (x—1)4  =  2(x+l)9  :  (x—  1)«, 
to  find  the  value  of  x.  i/24-l 


20.  Prove  that  a  :  b  =  c  :  d,  when 
(o+6-|-c+rf)  (a—  I—  c+d)  =  (a—  6-f-c—  d)  (a-j-&—  c—  <f). 


24 

i;; 


278  PERMUTATIONS   AND    COMBINATIONS. 


PERMUTATIONS  AND  COMBINATIONS. 


33*T.  The  Permutations  of  things  are  the  different  results 
obtained  by  placing  the  things  in  every  possible  order.  In  form- 
ing permutations,  all  of  the  given  elements,  or  a  part  only,  may  be 
taken  at  a  time ;  but  in  any  proposed  system,  the  different  results 
must  contain  the  same  number  of  things. 

Thus,  the  permutations  of  the  letters,  a,  b,  c,  taken  two  at  a  time, 
are 

a&,  5a,  ac,  ca,  be,  cb. 

The  permutations  of  the  same  letters  taken  all  at  a  time,  are 
cab,  acb,  abc,  cba,  bca,  bac. 

NOTE. — The  results  obtained  by  permuting  things,  where  less  than  all 
are  taken  at  a  time,  are  sometimes  called  variations  or  arrangements  ;  the 
word  permutations  would  then  be  restricted  to  the  case  in  which  all  the 
things  are  taken  at  a  time. 

336.  The  Combinations  of  things  are  the  different  collections 
that  can  be  formed  out  of  them,  without  regarding  the  order  in 
which  the  things  are  placed,  the  same  number  of  elements  entering 
into  all  the  results. 

Thus,  the  combinations  of  the  letters,  a,  6,  c,  taken  two  at  a  time, 
are 

abj  ac,  be. 

It  will  be  observed  that  if  the  letters  be  regarded  as  factors,  the 
combinations  which  may  be  formed  by  taking  n  at  a  time  will  con- 
stitute all  the  different  products  of  the  nth  degree,  of  which  the 
letters  are  capable. 

337^  To  find  the  number  of  permutations  of  n  things  taken  r 
at  a  time. 

Suppose  the  things  to  be  n  letters,  a,  b,  c,  d 

First : — If  we  take  each  of  the  n  letters  by  itself,  there  will  be  in 
every  case  n — 1  other  letters,  or  n — 1  reserved  letters. 

Now  if  to  each  of  the  n  letters  we  annex  each  of  the  reserved 
letters  successively,  we  -shall  form  all  the  permutations  with  two 


PERMUTATIONS   AND    COMBINATIONS.  279 

letters  each,  of  which  the  n  letters  are  susceptible.     But  we  shall 

obtain  every  time,  n — 1  results ;  thus, 

with  a,  aby         ac,         ad, ,         (n—  1  results) ; 

«    6,  6a,         be,         K...., 

"    c,  ca,         c6,         cd, ,    .*•      " 

«    d,  da,         db,        dc,   ..M 

etc.  etc.  etc. 

Now  since  there  are  n  letters,  each  to  be  combined  with  n — 1  re- 
served letters,  there  will  be  in  all  n(n — 1)  results.     That  is, 

The  number  of  permutations  of  n  letters  taken  two  at  a  time,  is 
n(n-l). 

/Second : — If  we  consider  each  of  the  permutations  of  the  n  let- 
ters with  two  in  a  set,  apart  from  the  other  letters,  there  will  be  in 
every  case  n — 2  reserved  letters.  Hence,  to  permute  the  n  letters 
with  three  in  a  set,  we  shall  have  n — 2  reserved  letters,  to  be  annexed 
successively  to  each  of  the  n(n — 1)  permutations  with  two  in  a  set, 
thus  forming  n(n — l)(w — 2)  new  results.  That  is, 

The  number  of  permutations  of  n  letters  taken  three  at  a  time,  is 
n(n-l)(n— 2). 

If  the  permutations  of  the  n  letters  taken  r — 1  at  a  time  were 
formed,  there  would  be  with  respect  to  each,  n — (r — 1),  or  n — r-j-1, 
reserved  letters.  And  we  might  conjecture  from  the  two  preceding 
cases,  that  the  number  of  permutations  of  n  letters  taken  r  at  a  time,  is 

n(»— 1)  (n— 2) ....  (n— r+1).  (4) 

or,  the  product  of  the  natural  .numbers  from  n  down  to  n — r-f-1, 
inclusive. 

This  may  be  demonstrated  in  a  general  manner,  as  follows  : 

Let  x  and  xr  represent  any  two  consecutive  numbers  less  than  n, 
so  that  x+1  =  xf.  (1) 

Let  P  represent  the  permutations  of  n  letters  taken  x  in  a  set, 
and  Pr  the  number  of  permutations  of  the  letters  taken  x-\-\  or 
x'  in  a  set. 

Now  if  we  consider  each  of  the  P  permutations  apart  from  the 
other  letters,  there  will  be  in  every  case  n — x  reserved  letters. 
Thus  we  have  n — x  reserved  letters  to  be  annexed  successively  to 
each  of  the  P  permutations,  in  order  to  form  the  P'  permutations 


280  PERMUTATIONS    AND    COMBINATIONS. 

with  x-j-1  or  x'  letters  in  a  set.  This  will  give  P(n — .r)  results  ; 
and  we  therefore  have 

P(n—x)  =  1*.  (2) 

Now  we  will  show  that  if,  according  to  the  law  already  enunciated, 

P  =  n(n— 1)  (n— 2) (n— z-f-1),  (3) 

then  the  value  of  P'  will  be  expressed  by  a  similar  formula.  For, 
multiply  both  members  of  equation  (3)  by  n — x,  and  equate  the 
result  with  the  second  member  of  (2) ;  we 'have 

*  P  =  ti(n^-l)  (n— 2)    ...  (n— x).  (4) 

But  from  equation  (1),  we  have 

x  =  x'—l. 
Substituting  this  value  of  x  in  equation  (4),  gives 

Pf  =  ??(?i— 1)  O— 2) (w_ o/4-l).         (5) 

Equations  (3)  and  (5)  are  similar  in  form.  Thus  we  have  shown 
that  if  formula  (J)  holds  when  the  letters  are  taken  :r  at  a  time,  it 
will  hold  when  the  letters  are  taken  x-{-\  at  a  time.  But  it  has 
been  proved  to  held  when  the  letters  are  taken  three  at  a  time  ; 
hence  it  holds  when  they  are  taken  four  at  a  time }  hence  also  it 
holds  when  they  are  taken  five  at  a  time,  and  so  on.  Thus  it  is 
true  universally. 

NOTE. — In  the  practical  application  of  formula  (A),  it  will  be  well  to 
remember  that  the  number  of  factors  is  equal  to  the  number  of  letters 
taken  in  a  set. 

338.  To  find  the  number  of  permutations  of  n  things  taken  all 
at  a  time,  put  r  —  n  in  formula  (A)  ;  the  result  will  be 

n(n— l)(n— 2)....l.  (#) 

That  is, 

The  number  of  permutations  of  n  tilings  taken  all  together  in  a 
set,  is  equal  to  the  continued  product  of  the  natural  numbers  from 
n  down  to  1,  inclusive. 

33O.  To  find  the  number  of  combinations  of  n  things  taken  r 
at  a  time. 

Let  Z  =  the  number  of  combinations  of  n  things,  taken  r  in  a  set ; 
P=  the  number  of  permutations  of  n  things,  taken  r  in  a  set : 
Pf  =  the  number  of  permutations  of  r  things,  taken  all  together 


PERMUTATIONS    AND    COMBINATIONS.  281 

Now  it  is  evident  that  all  of  the  P  permutations  can  be  obtained, 
by  subjecting  the  r  things  in  each  of  the  Z  combinations  to  all  the 
permutations  of  which  they  are  susceptible.  But  a  single  combina- 
tion of  r'  things  produces  P'  permutations,  taking  all  the  things  in 
a  set;  hence  the  Z  combinations  will  give  Zy.f  permutations,  and 
we  shall  therefore  have 


Whence,  Z=jj- 

But  by  (337),  P  =  ti(n—  !)(»—  2)  ....  (n—  r+1)  ; 

and  by  (338),  P'=  r(r—  l)(r—  2)  ....  1. 
Hence,  we  have 


-....- 
<r-l)(r-2)....l 
That  is, 

The  number  of  combination*  of  n  letters  taJten  r  at  a  time,  is 
equal  to  the  continued  product  of  the  natural  numbers  from  n  down 
to  n  —  r-|-l  inclusive,  divided  Ly  the  continued  product  of  the  nat- 
ural numbers  from  r  down  to  1  inclusive. 

•MO.  It  is  evident  that  for  every  combination  .of  r  things  which 
we  take  out  of  n  things,  there  will  be  left  a  combination  of  n  —  r 
things.  That  is,  every  possible  combination  containing  r  things, 
corresponds  to  a  combination  of  n  —  r  things  which  remain.  Hence, 

The  number  of  combinations  of  n  things  taken  r  at  a  time,  is 
equal  to  the  number  of  combinations  of  n  things  taken  n  —  r  at  a 
time. 

This  proposition  may  be  demonstrated  algebraically  as  follows  : 
Let  Z  represent  the  number  of  combinations  of  n  things  taken  r 
at  a  time,  and  Z'  the  number  of  combinations  of  n  things  taken  n  —  r 
at  a  time.     Let  it  be  observed  that  the  last  factor  in  the  numerator 
of  Z'  will  be  n—  (n—  r)-f  1  =  r+1.     Then 

n(n—  !)(«—  2)  ....  (n—  r-f  1)  ^ 


__ 


*—  2).... 


^ 
(n  —  r)(n  —  i  --  1)  ....  1 

24* 

-  17  .  .  .  -  / 


282  PERMUTATIONS   AND    COMBINATIONS. 

By  division  we  obtain 


Z'      n(nr-  1)  (7i—  2)... 
Whence,  Z=Z'. 

341.  7b  ^/iwc?  /or  «r/irt£  value  of  r,  tfie  number  of  combinations 
of  n  things  taken  T  at  a  time  is  the  greatest. 

Consider  r  as  a  varying  quantity,  being  at  first  unity,  and  chang- 
ing to  2,  3,  4,  ....  successively. 

Let  Z  represent  the  number  of  combinations  for  any  value  of  r, 
and  Z?  the  number  of  combinations  for  the  succeeding  value  of  r. 
We  have 

„       n(7i-l)O-2)  ....  (n—  r-f  2)  Q- 


And  if  in  this  equation  we  change  r  to  r-|-l,  the  result  must  be 
the  value  of  Z'  j  thus, 


_ 

V-l. 


Divide  (2)  by  (1),  observing  that  in  the  second  member  of  (2),  the 
factor  which  immediately  precedes  (n  —  r-f-1)  is  (n  —  r-j-2);  we 
have 


whence,  Z>  - 

\i  -pa./ 

Now  Z?  is  greater  or  less  than  Z,  according  as is  greater  or 

less  than  unity.     That  is,  when 


r+l" 

the  number  of  combinations  will  be  increased  by  giving  to  r  its 
succeeding  value ;  but  when 

n — r 

the  number  of  combinations  will  be  diminished  by  giving  to  r  its 
succeeding  value. 


^+>    /        r  \ 

?  * 

-  ^~ 

AND    COMBINATIONS.  283 


But  if 


And  if 


Hence,  that  value  of  r  which  will  give  the  greatest  number  of 
combinations,  must  not  be  less  than  —  —  ,  or  greater  than  —  -  --  [-1, 


,  .A     .„  r  f  Al     Al  . 

j  hence,  it  will  have  one  of  the  three  values, 


S.       'V  -  //'*/' 

t^/^v 


1st.    Suppose  n  even,.      Then  the  first  and  third  values  will  be 
fractionalj  and  therefore  impossible  for  r  ;  hence  in  this  case 


2d.  Suppose  n  odd.  Then  the  second  value  will  be  fractional, 
and  consequently  impossible  for  r  ;  hence,  in  this  case  r  must  have 
at  least  one  of  the  other  values.  We  will  show  that  it  may  have 
either  of  them.  For,  suppose 


' 

By  (34O),  the  number  of  combinations  will  be  the  same,  if 

n+l 


That  is,  when  n  is  odd,  the  greatest  number  of  combinations  will 

be  obtained  by  making  >• 

Ti—1  n-4-l 

r=—      or      r  =  — , 

the  two  values  of  r  giving  the  same  result. 

EXAMPLES  OF  PERMUTATIONS  AND  COMBINATIONS. 

1.  How  many  different  permutations  may  be  formed  of  10  letters 
taken  four  at  a  time  ?  Ans.  5040. 

2.  How  many  different  permutations  may  be  made  of  6  things 
taken  all  together  in  a  set  ?  Ans.  720. 


284  PERMUTATIONS   AND   COMBINATIONS. 

3.  How  many  different  permutations  may  be  made  of  10  things 
taken  all  together  ?  Ans.  3628800. 

4.  How  many  different  numbers  can  be  formed  with  the  five 
Arabic  characters,  4,  3,  2,  1,  0 ;  each  of  the  characters  occurring 
once,  and  only  once  in  each  number  ?  Ans.  90. 

5.  How  many  different  combinations  may  be  formed  of  8  things 
taken  4  at  a  time  ?  Ans.     70. 

6.  How  many  different  combinations  may  be  made  of  16  things 
taken  5  at  a  time  ?  Ans.  4368. 

7.  How  many  different  parties  of  6  men  each   can  be  formed 
from  a  company  of  20  men  ?  Ans.  38760. 

8.  In  how  many  different  ways  can  a  class  of  6  boys  be  placed 
in  line,  one  boy  being  denied  the  privilege  of  the  head  ? 

Ans.  60CT. 

9.  Find  the  greatest  number  of  different  products  that  can  be 
formed  with  the  prime  numbers  under  40,  the  products  being  all 
composed  of  the  same  number  of  factors.  Ans.  1716. 

10.  The  number  of  permutations  of  n  things  taken  5  at  a  time, 
is  equal  to  120  times  the  number  of  combinations  of  the  n  things 
taken  3  at  a  time ;  find  n.  Ans.  n  —  8. 

11.  At  a  certain  house  there  were  8  regular  boarders ;  and  one  of 
them  agreed  with  the  landlord  to  pay  $35  for  his  board  so  long  as 
he  could  select  from  the  company  different  parties,  equal  in  number, 
to  sit  each  for  one  day  on   a  certain  side  of  the  table.     At  what 
price  per  day  did  he  secure  his  board  ?  Ans.  $.50. 

12.  A  and  B  have  each  the  same  number  of  horses  j  and  A  can 
make  up  twice  as  many  different  teams  by  tj.kiug  3  horses  together, 
as  B  can  by  taking  2  together.     Required  the  number  of  horses 
that  each  has.  Ans.  8. 

13.  There  are  12  points  in  a  plane,  no  three  of  which  are  in  the 
same  straight  line  with  the  exception  of  five,  which  are  all  in  the 
same   straight   line.      How  many   different   straight  lines    can    be 
formed  by  joining  the  points.  Ans.  57. 


ARITHMETICAL   PROGRESSION.  285 


SECTION    VII. 

*- 

OF  SERIES. 

342.  A  Series  consists  of  a  number  of  terms  following  one 
another,  but  so  related  that  each  may  be  derived  from  one  or  more 
of  the  preceding,  by  a  fixed  law. 

A  scries  may  be  finite  or  infinite,  converging  or  diverging. 

343.  A  Finite  Series  is  one  which  by  its  law  of  development 
must  terminate,  or  have  only  a  finite  number  of  terms. 

344.  An  Infinite  Series  is  one  which  by  the  law  of  its  devel- 
opment can  never  terminate,  but  may  have  an  infinite  number  of 
terms. 

345.  A  Converging  Series  is  one  whose  successive  terms  con 
tinually  diminish  in  numerical  value. 

346.  A  Diverging  Series  is  one  whose  successive  terms  con- 
tinually increase  in  numerical  value. 

ARITHMETICAL  PROGRESSION. 

347.  An  Arithmetical  Progression  is  a  series  of  numbers  or 
quantities  increasing  or  decreasing  from  term  to  term  by  a  common 
difference. 

We  may  consider  the  common  difference  as  a  quantity  continually 
added,  in  the  algebraic  sense ;  hence,  it  will  be  positive  in  an  in- 
creasing series,  and  negative  in  a  decreasing  series.     Thus, 
1,     3,     5,     7,    9,.... 

is  an  increasing  arithmetical  progression,  in  which  the  common  dif- 
ference is  -f-2  j  and 

20,     18,     16,     14,     12,.... 

is  a  decreasing  arithmetical  progression,  in  which  the  common  dif- 
ference is  — 2. 


286  SERIES. 

348.  To  investigate  the  properties  of  an  arithmetical  progres- 
sion, we  may  suppose  the  series  to  terminate  ;  there  will  then  be 
five  parts  or  elements;  —  the  first  term,  the  last  term,  the  number  of 
terms,  the  common  difference,  and  the  sum  of  the  terms.  The  first 
term  and  last  term  are  called  the  extremes,  and  all  the  terms  between 
the  extremes  are  called  arithmetical  means. 

«$4:O.  In  an  Arithmetical  Progression,  the  last  term  is  equal  to 
the  first  term  plus  the  common  difference  multiplied  by  the  number 
of  terms  less  1. 

Let  a  denote  the  first  term,  I  the  last  term,  d  the  common  differ- 
ence, and  n  the  number  of  terms  j  then  the  series  will  be  represented 
thus  : 

a,  (a+rf),  (a+2d),  (a+3<f),.  .  ..*. 

And  we  perceive  that  in  every  term  the  coefficient  of  d  is  equal  to 
the  number  of  preceding  terms;  hence, 


in  which  d  is  positive  or  negative,  according  as  the  series  is  an  in- 
creasing or  a  decreasing  one. 

8*>O.  In  an  arithmetical  progression  the  sum  of  any  two  terms 
equidistant  from  the  extremes  is  equal  to  the  sum  of  the  extremes. 

Let  t  denote  a  term  of  the  series  which  has  r  terms  before  it,  and 
/'  a  term  which  has  r  terms  after  it  ;  then  the  terms,  t  and  t',  will 
be  equidistant  from  the  extremes.  Suppose  the  series  to  be  increas- 
ing ;  then  from  the  nature  of  the  series, 

*   =  a+rd  ;  (1) 

t'  =  l—rd',  (2) 

whence,  by  addition, 

t+t  =  a+L 

«£51.  The  sum  of  the  terms  of  an  Arithmetical  Progression  is 
'qua!  lo  one  half  the  sum  of  the  two  extremes,  multiplied  by  the  num- 
ber of  terms. 

Represent  the  sum  of  the  series  by  S,  then  we  have 

S  =  a-f(a+d)+(a+2d)-f  .  .  .  .  +1.  (1) 

By  writing  the  series  in  the  reverse  order,  we  have  also 

S  =  **—  d)+J-t2d+.  ...+«.  (2) 


ARITHMETICAL    PROGRESSION.  287 

Therefore,  by  addition, 


Now  equation  (3)  expresses  the  sum  of  n  terms,  each  equal  to 
(a-|~0 ;  hence, 

and  dividing  by  2,  we  obtain  the  formula, 

n 


.  Jb  insert  any  number  of  arithmetical  means  between  two 
given  terms. 

Let  nf  denote  the  number  of  means  to  be  inserted.  Then  the 
number  of  terms  in  the  completed  series  will  be  n'-f-2  j  and  we 
shall  have  n  =  w'-f-2. 

This  value  of  n  substituted  in  formula  (4),  (34L0),  gives 


,        l-a 
whence,  =  ^7-p1.  ((7) 

Having  the  common  difference,  the  means  are  readily  obtained. 


APPLICATION  OF   THE  FORMULAS. 

353.  The  two  formulas, 

I  =  a+(n—l)d,  (A) 


contain  in  all  five  quantities,  a,  /,  n}  rf,  S,  four  of  which  enter  eacK 
equation.  Now  if  any  three  of  these  quantities  be  given,  the  other 
two  may  be  found  ;  for,  if  the  values  of  the  three  given  quantities 
be  substituted  in  the  formulas,  there  will  result  two  equations  con- 
taining only  two  unknown  quantities. 

1.  The  first  term  of  an  arithmetical  series  is  5,  the  common  dif- 
ference 3,  and  the  number  of  terms  24.  Find  the  last  term,  and 
*the  sum  of  the  series. 

We  have  given,  a  —  5,  tf=3,w  =  24; 

hence,  by  formula  (A),      I  —  5+(24—  1)3  =    74  ) 
and  bj  formula  (B),  S  =  ^  (5^.74^    _  943  } 


288  SEEIES. 

2.  Given  a  =  15,  d  =  —2,  and  S  =  60,  to  find  the  number  of 
terms. 

Substituting  the  given  values  in  (A)  and  (Z»),  we  have 

1=  15— 2(7i-l),  (1) 

60  =  ?  (15+0;  (2) 

whence,  from  (1),  I  =  17— 2n, 

120— Ion 

and  from  (2),  I  = 

n 

120— 15» 


.     =n-2», 

120— 15?i  =47>i— 2na 
n«_16n=— 60, 


n  =  10  or  6. 

Both  values  of  n  arc  possible ;  for  there  are  two  scries  answering 
to  the  given  conditions,  one  having  6  terms,  and  the  other  10 ; 
these  are 

15,    13,    11,    9,    7,    5,    3,    1,    -1,    -3, 
and  15,     13,     11,    9,     7,    5. 

The  sum  of  cither  scries  is  60. 

EXAMPLES  FOE  PRACTICE. 

1.  The  first  term  of  an  arithmetical  scries  is  7,  the  common 
difference  3,  and  the  number  of  terms  36  ;  find  the  last  term. 

Ana.  112. 

2.  The  first  term  of  an  arithmetical  series  is  275,. the  last  term 
5,  and  the  number  of  terms  46 ;  required  the  sum  of  the  terms. 

Ans.  6440. 

3.  The  sum  of  an  arithmetical  series  is  156,  the  number  of 
terms  8,  and  the  common  difference  5.    Required  the  two  extremes. 

4.  Find  the  sum  of  the  terms  in  an  arithmetical  progression, 
knowing  that  the  first  term  is  1,  the  common  difference  f,  and  the 
number  of  terms  101.  Ans.  2626. 


ARITHMETICAL    PROGRESSION.  289 

5.  Required  to  find  four  arithmetical  means  between  7  and  37. 

Ans.  13,  19,  25,  31. 

6.  The  first  term  of  an  arithmetical  series  is  3,  the  number  of 
terms  60,  and  the  sum  of  the  terms  3720;  required  the  common 
difference,  and  the  last  term.  Ans.  *d=  2,  I  -_—  121. 

-"*  7.  What  will  be  the  sum  of  the  series  if  9  arithmetical  means  bo 
inserted  between  9  and  109  1  Ans,  649. 

8.  If  three  arithmetical  means  be  inserted  between  ±    and  A, 
what  will  be  the  common  difference?       /  Am.  5'?. 

9.  What  debt  can  be  discharged  in  a'year  by  paying  1  cent  the 
first  day,  3  cents  the  second,  5  cents  the  third,  and  so  on,  increasing 
the  payment  each  day  by  2  cents  ?      Ans.  1332  dollars  25  cents. 

10.  A  footman  travels  the  first  day  20  miles,  23  the  second,  26 
the  third,  and  so  on,  increasing  the  distance  each  day  3  miles.    How 
many  da}*s  must  he  travel  at  this  rate  to  go  43£  miles  ?   Ans.  12. 

11.  Find  the  sum  of  n  terms  of  the  progression  of  1,  2,  3,  4, 
5'6'  .....  -  An*.  S  = 


12.  Find  the  sum  of  n  terms  of  the  progression  1,  3,  5,  7,  ..... 

Ans.  S  =  n\ 

13.  The  sum  of  the  terms  of  an  arithmetical  series  is  950,  the 
common  difference  is  3,  and  the  number  of  terms  25.     What  is  the 
first  term  ?  Ans.  2. 

>•  14.  A  man  bought  a  certain  number  of  acres  cf  land,  paying  for 
the  first  $  j  ;  for  the  second,  $:j  ;  and  so  on.  When  he  came  to 
settle  he  had  to  pay  $3775.  How  many  acres  did  he  purchase  ? 

Ans.  150  acres. 

1  15.  The  14th,  134th,  and  last  terms  of  an  arithmetical  progres- 
sion are  66,  666,  and  6666,  respectively.  Ilequired  the  number  of 
terms.  Ans.  1334. 


(a  -££  66  6  £  -  (>(, 


.  f-0    -4-  t  1L0  V  /  ^    i.  /  3   ? 

25  T 


290 


SERIES. 


THE   TEN    CASES. 

I.    Given  any  three,  of  the  quantities,  a,  1,  n,  d,  S,  to  find 
the  other  two. 

This  problem  will  present  ten  cases,  each  giving  rise  to  two  for- 
mulas, making  in  all  twenty  different  formulas,  or  four  values  for 
each  letter.  The  results  in  each  case  may  be  obtained  directly  from 
the  two  fundamental  equations,  or  those  of  any  particular  case  my 
be  derived  from  some  preceding  case,  as  is  most  convenient.  The 
whole  wril  be  left  as  an  exercise  for  the  student. 


Given. 

a,  d',  n 


I,  d,  n 


«,  n,  I 


d,n,S 


l,n,S 


a,  d,l 


a,l,S 


l,d,S 


To  Find. 


a,  8 


d,S 


a,  I 


1,   S 


n,  d 


n,a    A  _ 


Formulas. 


=  x«+o- 


2S—n(n—V}d 
2^ 


w—  !>/ 


n(n— 


/       2S 

Z  = a. 

n 


~  T~\~ 

w(«  —  1) 


S  = 


2^—  (/-j-a)  ' 


-',  J=a+(n-lX 


AKITHMETICAL    PROGRESSION.  291 

PROBLEMS  IN  ARITHMETICAL  PROGRESSION 

10   WHICH   THE   FORMULAS   DO    NOT    IMMEDIATELY   APPLY. 


When  in  the  conditions  of  a  problem  no  three  of  the 
five  parts,  a,  /,  ?i,  d,  S,  are  directly  given,  the  general  formulas  will 
not  directly  apply.  *It  is  usually  necessary  in  such  instances  to 
represent  the  several  terms  of  the  series  by  means  of  two  or  more 
unknown  quantities  ;  and  for  this  purpose  there  are  two  methods 
of  notation. 

1st.  Let  x  denote  the  first  term  and  y  the  common  difference  ; 
thus, 


This  method  of  notation,  however,  is  seldom  the  most  expedient. 

2d.  When  the  number  of  terms  is  odd,  denote  the  middle  term 
by  .r,  and  the  common  difference  by  y  ;  then  we  shall  have, 
for  three  terms,  (x  —  y),  x,  (x-\-y)  ; 

for  five  terras,         (x—2y),  (x—  y),  x,  (x+y),  (*+2y). 

And  when  the  number  of  terms  is  even,  represent  the  two  middle 
terms  by  x  —  y  and  x-\-y  respectively,  2y  being  the  common  differ- 
ence ;  thus, 

(x-3y),  (x-y),  (x+y),  (*+%)• 

The  advantage  of  the  second  method  is,  that  the  sum  of  all  the 
terms,  or  the  sum  and  difference  of  two  terms  equidistant  from  the 
extremes,  will  each  contain  but  a  single  unknown  quantity. 

1  .  There  are  three  numbers  in  arithmetical  progression  ;  the  sum 
of  these  numbers  is  18,  and  the  sum  of  their  squares  is  158. 
What  are  the  numbers?  Ans.  1,  6,  11. 

2.  There  are  five  numbers  in  arithmetical  progression  ;  their  sum 
/      is  65,  and  the  sum  of  their  squares  1005.     What  are  the  numbers  ? 

Ans.  5,9,  13,  17,21. 

3.  Tt  is  required  to  find  four  numbers  in  arithmetical  progression, 
such  that  their  common  difference  shall  be  4,  and  their  continued 
product  176985.  Ans.  15,  19,  23,  27. 

\~t         r'1"          ^'7-f'-~ 


292 


SERIES. 


4.  There  are  four  numbers  in  arithmetical  progression  ;  the  sum 
of  the  extremes  is  8,  and  the  product  of  the  means  15.     What  are 
the  numbers?  Ans.  1,  3,  5,  7. 

5.  A  person  starts  from  a  certain  place  and  goes  1  mile  the  first 
day,  2  the  second,  3  the  third,  and  so  on ,  in  six  days  after,  another 
sets  out  from  the  same  place  in  pursuit,  and  travels  uniformly  15 
miles  a  day.     How  many  days  after  the  second  starts  before  they 
are  together  ?  <l  ^    Ans.  3  days,  and  14  days. 

NOTE. — Reconcile  these  two  values. 

6.  A  man  has  borrowed  $60.     What  sum  shall  he  pay  daily  to 
cancel  the  debt  in  60  days  j   interest  being  allowed  on  the  sum 
borrowed  for  the  whole  time,  and  on  each  payment  from  the  time  it 
is  made  to  the  end  of  the  60  days,  at  the  rate,  of  10  per  cent,  for 
12  months  of  30  days  each  ?  Ans.  $1 7§|¥. 

7.  There  are  four  numbers  in  arithmetical  progression  ;  the  sum 
of  the  squares  of  the  extremes  is  657  and  the  sum  of  the  squares 
of  the  means  is  61.     Required  the  numbers.         Ans.  4,  5,  6,  7. 

8.  The  sum   of  four  numbers  in  arithmetical  progression  is  24, 
and  their  continued  product  is  945  ;  what  are  the  numbers  ? 

Ans.  3,  5,  7,  9. 

9.  A  certain  number  consists  of  three  digits,  which  are  in  arith- 
metical progression  ;  if  the  number  be  divided  by  the  sum  of  its 
digits  the  quotient  will  be  26,  and  if  198  be  added   to  the  number 
its  digits  will  be  inverted.     What  is  the  number  ?          Ans.  234. 

••r  /}  10.  From  two  towns  which  were  102  miles  apart,  two  persons, 

i  jjA  and  B,  set  out  to  meet  each  other ;  A  went  3  miles  the  first  day, 

5  the  next,  7  the  next,  and  so  on ;  B  went  4  miles  the  first  day,  6 

the  next,  8  the  next,  and  so  on.     In  how  many  days  did  they  meet? 

Aiis.  6. 

11.  A  quantity  of  corn  is  to  be  divided  among  21  persons,  and  is 
calculated  to  last  a  certain  time  if  each  of  them  receive  a  peck  every 
week ;  during  the  distribution  it  is  found  that  one  person  dies  at 
the  end  of  every  week,  and  then  the  corn  lasts  twice  as  long  as  was 
expected.  Required  to  find  the  quantity  of  corn. 

Am.  231  pecks. 


GEOMETRICAL    PROGRESSION.  293 

GEOMETRICAL  PROGRESSION. 

356.  A  Geometrical  Progression   is   a   series  of  quantities, 
each  of  which  is  equal  to  the  preceding  one  multiplied  by  a  con- 
stant factor. 

357.  The  constant  factor  is  called  the  ratio ;  and  if  the  first 
term  is  positive,  the  progression  will  be  an  increasing,  or  a  decreas- 
ing series,  according  as  the  ratio  is  greater  or  less,  than  unity. 

Thus,  2,     6,     18,     54,     162,   .... 

is  an  increasing  geometrical  series,  in  which  the  ratio,  is  3  ;  and 

81,     27,     9,     3,     1,     I,     4 

is  a  decreasing  geometrical  series,  in  which  the  ratio  is  -|. 

358.  When  a  geometrical  progression  is  supposed  to  terminate, 
the  first  and  last  terms  are  called  extreme*,  and  all  the  terms  between 
the  first  and  last  are  called  geometrical  means. 

35O.    To  find  the  last  term  of  a  geometrical  progression. 
Let  a  denote  the  first  term,  r  the  ratio,  I  the  last  term,  and  n  the 
number  of  terms.     Then  the  series  will  be  represented  thus : 

a,     ar,     ar9,     ar8,    ....    I. 

Now  we  perceive  that  in  any  term  the  exponent  of  r  is  equal  to  th<» 
number  of  preceding  terms.     Hence,  we  shall  have 

r<£=-  *^"*  /  =  ar»-i .  (A)  + 

36O.    To  find  the  sum  of  the  terms  in  a  geometrical  progression, 
Denote  the  sum  of  the  scries  by  S]  then 

S  =  a-\-ar-\-ar*+  ar* -far"-1,  (1) 

and  rS  =         ar-\-ar*+ar* _|_ar«-i  _|_ar«.    (2) 

Hence  by  subtraction,  remembering  that  arn  =  rl, 

rS—S  =  ar»—a,  (3) 

or,  rS—  S  =  rl— a.  (4) 

Thus  we  obtain  two  expressions  for  S,  as  follows : 


(*) 

T J. 

25* 


294 


SERIES. 


361.  To  find  the  sum  of  a  decreasing  geometrical  series,  when 
the  number  of  terms  is  infinite. 

By  changing  signs  in  the  numerator  and  denominator,  equation  (#) 
may  be  written 

e_q(l_rn) 

l-r    • 

Now  suppose  r  less  than  unity ;  then  the  larger  n  is,  the  smaller 
will  rn  be ;  and  by  making  n  large  enough,  rn  may  be  made  less  than 
any  assignable  quantity,  or  zero.  Hence,  if  the  number  of  terms  is 
infinite,  rn  may  be  neglected  in  comparison  with  unity ;  and  we  shall 
have  as  the  formula  for  an  infinite  series, 


363*  To  insert  a  given  number  of  geometrical  means  between 
two  given  quantities. 

Let  n'  denote  the  number  of  means  to  be  inserted ;  then  the 
whole  series  will  consist  of  nf-{-2  terms.  Hence,  putting  n  =  n'-\-2 
in  equation  (^),  we  have 

Whence  we  obtain, 

r  = 

Having  found  the  ratio,  the  required  means  may  be  obtained. 
*  363.  Since  the  terms  of  a  geometrical  series,  taken  consecu- 
tively, have  the  same  ratio  one  to  another,  it  follows  that  they  are  in 
continued  proportion  ;  (337).     Hence, 

1st.  When  three  terms  are  in  geometrical  progression,  the  product 
of  the  extremes  is  equal  to  the  square  of  the  mean. 

2d.  When  four  terms  are  in  geometrical  progression,  the  product 
of  the  means  is  equal  to  the  product  of  the  extremes. 

APPLICATION   OF   THE   FORMULAS. 

364.  The  two  primitive  equations, 


contain  the  five  quantities,  a,  r,  ?,  n,  S,  any  three  of  which  being 

- 


GEOMETRICAL    PROGRESSION.  2U5 

given,  the  other  two  may  be  found  ;  for,  by  substitution  of  the 
given  values,  the  result  will  always  be  two  equations  involving  but 
two  unknown  quantities. 

In  this  general  problem  there  will  be  ten  cases,  as  in  the  corres- 
ponding problem  of  Arithmetical  Progression.  ^Ve  can  not,  howev- 
er, obtain  a  solution  of  all  the  cases,  by  simple  or  quadratic  equations. 

1st.  The  quantity  n  enters  the  two  equations  only  as  an  exponent, 
and  its  value  can  not  be  obtained  by  the  common  methods  of  reduc- 
ing an  equation.  The  process  involves  the  principle  of  logarithms, 
and  will  be  presented  in  its  proper  place. 

2d.  The  quantity  r  is  affected  by  an  exponent  in  both  equations; 
and  its  value  must  be  obtained  by  extracting  the  (n  —  Y)th  or  the 
nth  root  of  a  quantity.  When  n  is  not  large,  r  can  readily  be 
found  by  inspection  or  trial. 

3d.  The  values  of  a,  I,  and  8  will  be  found  by  means  of  simple 
equations,  as  in  Arithmetical  Progression. 

1.  The  first  term  of  a  geometrical  progression  is  3,  and  the  ratio 
2  ;  find  the  12th  term,  and  the  sum  of  the  series. 

We  have  given 

a  =  3,       r  =  2,       n  =  12. 
Whence  by  formulas  (A)  and  (£), 

f  =3X2"  =  3x2048  =    6144] 

3(9"  _  ";n  V     Ans. 

S  =    (~9_l  }  =  3X4095  =  12285  f 

2.  The  sum  of  a  geometrical  progression  is  1820,  the  number  of 
terms  6,  and  the  ratio  3  j  find  the  first  term,  and  the  last  term. 

We  have  given, 

S  =  1820,      n  =  6,      r  =  3. 
By  formula  (J5), 


a  =  5,  first  term. 
Then  by  formula  (JL), 

/  =  5X36  =  1215,  last  term. 


296  SERIES. 

3.  It  is  required  to  find  3  geometrical  means  between  6  and  486. 
By  formula  (Z>),  we  have 

r  =  Vigi  =  Vyi  =  3. 
Therefore,  the  series  is  6,  18,  54,  162,  486,  Ans. 

4.  Find  the  sum  of  the  series  G,  2,  §,}...  .to  infinity. 
We  have  given,    a  =  6,    r  •=.  \;  hence,  by  formula  ((7), 

ffs-p-SB*,  Am. 

5.  Find  the  exact  value  of  the  decimal  .454545. . .  .to  infinity. 
This  is  a  circulating  decimal,  and  may  be  expressed  thus  : 

45          45  45 

UK)  ~*~  10000  ~*~  1000000  ~1 

In  all  such  cases,  the  repetend,  taken  with  its  local  value,  will  bo 
the  first  term  of  a  geometrical  series,  of  which  the  ratio  will  be  10 
or  some  power  of  10.  In  the  present  example  we  have 

0  =  rfo'     r  =  m'9  hcncc' 

e_  45       (i          l   \_  45        100_  5 

"lOO^V1    ~  100/— 100  X  99  -If    ***' 

6.  Find  the  value  of  1 1 — . .  4-  •  • « •  to  infinity. 

a       a       a 

We  have    a  =  1,    r  = ',  Lcnco, 


]=— —  ,    Ans. 
x          «+x 


L-V-  %i  — 

£y       ^  -w^ 

A  -r  — 
EXAMPLES  FOR   PRACTICE. 

1.  Find  the  sum  of  9  terms  of  the  series  1,  2,  4,  8,  .... 

Ans.  511. 

2.  Find  the  8th  term  of  the  progression  2,  6,  18,  54, 

Ans.  4374. 
3    Find  the  sum  of  10  terms  of  the  series  1,  |,  $,  28f , 


j   a 

GEOMETRICAL    PROGRESSION.  297 

4.  Required  two  geometrical  means  between  24  and  192. 

.4ns.  48,  96. 

5.  Required  7  geometrical  means  between  3  and  768. 

Am.  6,  12,  24,  48,  96,  192, 

G.  Find  the  value  of  1  -f-  }.+  T9g  +  f  f  +  .  .*  ,  to  infinity. 

AM.  4. 

7.  Find  the  value  of  f  +  1  +  f  +  T&  -f  ----  to  infinity. 

-4ns.  4£. 

8.  Find  the  value  of  5  -j-  |  -j-  f  -f-  2\  -f-  .  .  .  .  to  infinity. 

Ana.  7f 

9.  Find  the  value  of  the  decimal  .323232.  .  .  .  to  infinity. 

Aus.  -§§. 

10.  Find  the  value  of  the  decimal  .212121  ____  to  infinity. 

Ans.  sV 

11.  Find  the  value  of  4  —  j  +  J  —  ^  -f  •&  —  ----  to  infinity. 


12.  Find  the  value  of  \  —  ^  -f-  3^T  —  c£5  -f  ----  to  infinity. 


a;      x*      x* 
13.  Find  the  value  of  1  -\  ---  f-  -,  -j-  ^  -f  ----  to  infinity. 


a 

Am.   - 
a  —  x 


1          CC*          X*         X* 

14.  Find  the  value  of  ---  ,  -j-  -  -.  --  ;  +  •  •  •  »*°  infinity. 
a      a*   '    a*      aT 


a'-J-x3 

15.  The  sum  of  a  geometrical  series  is  1785,  the  ratio  2,  and  the 
number  of  terms  8  ;  find  the  first  term.  Ans.  7. 

16.  The  sum  of  a  geometrical  series  is  7812,  the  ratio  5,  and  the 
number  of  terms  6  ;  find  the  last  term.  Ans.  6250. 

17.  The  first  term  of  a  geometrical  series  is  5,  the  last  term  1215, 
and  the  number  of  terms  6.     What  is  the  ratio  ?  Ans.  3. 

18.  A  man  purchased  a  house  with  ten  doors,  giving  $1  for  the 
first  door,  $2  for  the  second,  $4  for  the  third,  and  so  on.     What 
did  the  house  cost  him  ?  Ans.  $1023. 


298  SERIES. 

PROBLEMS  IN  GEOMETRICAL  PROGRESSION 

TO    WHICH    THE    FORMULAS    DO    NOT   IMMEDIATELY    APPLY. 

9 

3G5.  The  terms  of  a  geometrical  progression  are  represented 
in  a  general  manner  as  follows  : 

x,     xy,     xy*,     xy\  ---- 

In  the  solution  of  problems,  however,  the  following  notation  is  gen- 
erally preferable  : 

1st.  When  the  number  of  terms  is  odd,  the  series  may  be  repre- 
sented thus  : 

V-^  x\      xy,       y*  ; 

*  *'        .  \j 

-'  x>  x^  v>  -*> 

2d.  When  the  number  of  terms  is  even,  the  series  may  be  ex- 
pressed thus  : 

*  *  y* 

r~£  p   f^*« 


We  may  also  represent  three  terms  as  follows  : 

y*  ,/  — 

r  *  ~fe  x'        IJ*    y° 

1.  'The  sum  of  three  numbers  in  geometrical  progression  is  26, 
and  the  sum  of  their  squares  364.     What  are  the  numbers  1 

Let  the  numbers  be  denoted  by  x,  V  '  xy,  y. 
Then  x-\-T/xy+y  =    26  =  a,  (1) 

and  «3-|-xy-t-y2  =  364  =  b.  (2) 

Transposing  V  xy  in  (1),  squaring  and  reducing, 

*a+^-f-y  =  a3—  2al/^.  (3) 

From  (2)  and  (3),  a8  —  2al/xy  =  b  ; 

/—        a*—b 
whence,  V  xy  •=.  ——  —  =  6. 

From  (1)  and  (2)  x  =  2,  and  y  =  18. 

Hence,  the  numbers  are,  2,  6,  18,     Ans. 


GEOMETRICAL   PROGRESSION.  299 

2.  The  sum  of  four  numbers  in  geometrical  progression  is  15   or 
a,  and  the  sum  of  their  squares  85  or  b.     What  are  the  numbers  ? 
Taking  the  proper  notation  for  an  even  number  of  terms,  we  have  " 


and,  ^+x'+y>+^==6.>  "  (2) 

ty 

Assume      &-\-y  =  s,  and  xy=p-,  then  by  (3O1), 

cca-f  if  =  s*—2pi         ;  x*+y*.  =  s8—  3«p. 
Substituting  the  values  of  (z-fy)  and  O2-fy2),  in  (1)  and  (3), 


Squaring  (3),  and  then  transposing  2;cy,  or  2p, 


whence,  from  (4)  and  (5),     («—  s)a—  2p  =  6—  s*+2p  ; 

or,  a9—  2as-|72s2—  4p  ==  b.      ,       (6) 

Clearing  (o)  of  fractions,  and  putting  xy  =p  in  second  member, 

x*  -\~y*  =  a^  —  1>*  j      or>     s*  —  3sp  =  ap  —  ps  j 
whence,  p  =  -_i_.  (7) 

Substituting  this  value  of  p  in  (6),  and  reducing,  we  have 
a»_2a.s2  =  ab+2bs  ; 


or,  as*-}-  5s  =  —  (a*  —  6). 

Restoring  the  numerical  values  of  a  and  6, 

15s'+85s  =  70x15,- 
whence,  *  =  6. 

Substituting  the  values  of  a  and  s  in  (9),  and  we  obtain 

P  =  8; 

that  is,  x~\~y  =  6,         xy  =  8  ; 

whence,  x  =.  2,  y  =  4. 

Therefore,  the  required  numbers  are  1,  2,  4,  8,  Ans. 


SERIES. 

3.  There  are  three  numbers   in  geometrical  progression;    their 
sum  is  21,  and  the  sum  of  their  squares  is  189.     Find  the  numbers. 

Ans.  3,  6,  12. 

4.  Divide  the  number  210  into  three  parts,  so  that  the  last  shall 
exceed  the  first  by  90,  and  the  parts  be  in  geometrical  progression. 

*  ,  *y  f  Y  y  *-  Ans.  30,  60,  and  120. 

5.  The  sum  of  four  numbers  in  geometrical  progression  is  30  ; 
and  the  last  term  divided  by  the  sum  of   the  mean  terms  is   1^. 
What  are  the  numbers  ?   K»  fy-  *^/~*YJ  Ans.  2,  4,  8,  and  16. 

6.  The  sum  of  the  first  and  third  of  four  numbers  in  geometrical 
progression  is  148,  and  the  sum  of  the  second  and  fourth  is  888. 
What  are  the  numbers  ?X/Yy'Vy**y?te.  4,  24,  144,  and  864. 

7.  It  is  required  to  find  three  numbers  in  geometrical  progression, 
such  that  their  sum  shall  be  14,  and  the  sum  of  their  squares  84. 

Ans.  2,  4,  and  8. 

S.  There  are  four  numbers  in  geometrical  progression,  the  sec- 
ond of  which  is  less  than  the  fourth  by  24  ;  and  the  sum  of  the 
extremes  is  to  the  sum  of  the  means  as  7  to  3.  What  are  the  num- 
bers? Y"  >y  •  *  y  %  *y  3  Ans.  1,  3,  9,  and  27.  _, 

9.  There  are  three  numbers  in  geometrical  progression  ;  the  sum 
of  the  first  and  second  is  20,  and  the  difference  of  the  second  and 
third  is  30.  What  are  the  numbers  ?  X,*y*  *y3lns.  5,  15,  45. 

10.  The  continued  product  of  three  numbers  in  geometrical  pro- 
gression is  216,  and  the  sum  of  the  squares  of  the  extremes  is  328. 
What  are  the  numbers  ?  }L  ,  Y-y  ,  *  y  *"        Ans.  2,  6,  18. 

11.  The  sum  of  three  numbers  in  geometrical  progression  is  13, 
and  the  sum  of  the  extremes  being  multiplied  by  the  mean,  the 
product  is  30.     What  are  the  numbers^—^        Ans.  1,  3,  and  9. 

12.  There  are   three  numbers  in  geometrical  progression  ;  their 
continued  product  is  64,  and  the  sum  of  their  cubes  is  584.     What 
are  the  numbers  ?  /  /  /*/  *    *"  Ans.  2,  4,  8. 


13.  There  are  three  numbers  in  geometrical  progression;  their 
continued  product  is  1,  and  the  difference  of  the  first  and  second  is 


IDENTICAL   EQUATIONS.  301 

to  the  difference  of  the  second  and  third  as  5  to  one.     What  aro 
the  numbers?  '+*  *     ^  Ans'        l    5' 


of  120  dollars  was  divided  between  four  persons  in 

fJL0  such  a  manner  that  the  shares  were  in  arithmetical  progression  ;  if 
the  second  and  third  had  each  received  12  dollars  less,  and  the 
fourth  24  dollars  more,  the  shares  would  have  been  in  geometrical 

""^''p^gfesiJn/^Jind  the  shares.         Ans.  $3,  $21,  $39,  and  $57. 
/*-*:.  <W-°?.-/*  +  7 

lo.  There  are  three  numbers  in  geometrical  progression,  whose 

sum  is  31,  and  the  sum  of  the  first  and  last  is  26.     What  are  the 
numbers?         V-,</Vy,  y  Ans.  1,  5,  and  25. 

>/       16.  The  sum  of  six  numbers  in  geometrical  progression  is  189, 
and  the  sum  of  the  second  and  fifth  is  54.     What  are  the  numbers  ? 
*/  *yr*Y%Xy2f  *-</*,  +>*f'~          Am.  3,  6,  12,  24,  48,  and  96. 

17.  The  sum  of  six    numbers  in    geometrical  progression  is  189, 
and  the  sum  of  the  two  means  is  36.     What  are  the  numbers  ? 

AUK.  3,  6,  12,  24,  48,  and  96. 

18.  A  man  borrowed  p  dollars  ;  what  sum  must  he  pay  yearly  in 
order  to  cancel  the  debt  in  n  years,   interest  being  allowed  on  the 
unpaid  parts  of  the  principal  at  r  cents  per  annum  on  a  dollar  ? 


IDENTICAL  EQUATIONS. 

An  Identical  Equation  is  one  in  which  the  two  members 
are  either  the  same  algebraic  expression,  or  the  one  member  is  merely 
another  form  for  the  other.  In  every  case,  either  the  one  member 
may  be  reduced  to  the  other  directly,  or  the  two  members  may  be 
reduced  to  some  expression  different  from  either,  from  which  both 
members  may  be  supposed  to  originate.  Thus, 
ax-{-b  =  ax-\-b, 


,  ,  __    _    __         _ 

~l+x  —  x  ~  *• 
are  identical  equations.     In  the  first,  tne  two  members  have  exactly 
26 


302  SERIES. 

the  same  form.     In  "the  second,  the  second  member  may  be  reduced 
to  the  form  of  the  first,  by  performing  the  multiplication   indicated. 

In  the-third,  each  member  may  be  reduced  to  the  fraction,  ---  . 

1-j-x 

367.  There  are  certain  properties  of  identical  equations,  which 
are  of  great  importance  in  the  further  treatment  of   series,  and  iu 
the  general  theory  of  equations. 

In  order  to  investigate  these  properties,  let  us  first  consider  what 
any  term  containing  the  variable  x,  as  axn,  will  become  when  x  =  0, 
under  the  various  conditions  of  the  exponent. 

1  —  Suppose  n  to  be  positive  ;  then  if  x  •=.  0,  we  have 
ax*  =  a  '  0*  ==  0. 

2.  —  Suppose  n  to  be  negative  ;  then  if  x  =  0,  we  have 

a        a 

ax    =^=o=GO- 

3.  —  Suppose  n  to  be  nothing  or  zero  ;  then  if  x  =  0,  we  have 

ax*  =  a  -  ()•  =  a  •  1  =  a.       Q*  *   -j     I 

368.  We  are  now  prepared  to  demonstrate  the  following  propo- 
sitions : 

I.  An  identical  equation  is  satisfied  for  any  value  whatever  of  the 
unknown  quantity. 

The  truth  of  this  proposition  follows  directly  from  the  definition 
of  an  identical  equation.  It  is  implied  in  all  algebraic  transforma- 
tions, that  the  value  of  a  function  is  not  changed  by  changing  its 
form,  whatever  quantities  the  symbols  represent.  Hence,  if  the 
two  members  of  an  equation  are  the  same  in  form,  or  reducible  to 
the  same  expression,  they  must  be  equal,  whatever  value  be  substi- 
tuted for  the  unknown  quantity. 

To  illustrate  this  principle,  we  will  take  the  following  identical 
equation, 


where  the  form  is  such  that  the  identity  of  the  two  members  is  not 
apparent  from  inspection. 


IDENTICAL    EQUATIONS.  303 

If  in  this  equation  we  make  x  equal  to  1,  2,  3,  4,  5,  etc.  succes- 
sively, we  shall  have, 

{4+1+1}' =  2(16+1+  1}, 
{1+1+0}' =  2}  1+1+0}, 
{0+1+1}*:=  2{  0+1+*!}, 
{ 1+1+4  f  =  2{  1+1+16}, 
{ 4+1+9  }'  =  2{  16+1+81  f,  etc., 
every  result  being  a  true  equation. 

II.  Conversely : — Every  equation  which  is  satisfied  for  any  value 
whatever  of  the  unknown  quantity,  is  an  identical  equation. 

Suppose  the  given  equation  to  be  cleared  of  fractions,  and  each 
member  arranged  according  to  the  ascending  powers  of  the  unknown 
quantity.  Then  the  equation  may  be  represented  thus  : 

Atf+£a*+  <7xe+  . . . .  =  4V+.BV+  CV'+  ....  (D 

in  which,  by  hypothesis,  we  have 

a  <  b  <  c   .  . . ,  and  a '  <  b'  <  c' 

It  is  implied,  also,  that  the  coefficients,  A,  B,  (7,  etc.,  and  A',  B'j  (7, 
etc.,  are  all  finite  quantities  greater  than  zero,  and  independent  of*', 
and  the  number  of  terms  may  be  limited  or  unlimited. 
Divide  both  members  of  equation  (1)  by  xa'}  we  have 
A+£xb-a+  OB—+ . . .  .  =  A'xu'-~+£'x*'—+  C'x"—+ . . . ,    (2) 
in  which  the  exponents,  b — a,  c — a,  etc.,  in  the  first  member,  are  all 

positive,  because  a  <  b  <  c 

Now  by  hypothesis,  the  given  equation  is  true  for  all  values  of 
x  j  hence  every  modification  of  it  will  be  true  for  all  values  of  x. 
Make  x  =  0 ;  then  in  the  first  member  of  equation  (2),  every  term 
after  the  first  will  reduce  to  zero,  (307,  1),  and  we  shall  have 

A  =  A  V— +  J?'**'— +  C'xe'-*+  ....  (3) 

Now  since                                         a'  <  br  <  c' . . . . , 
we  must  have    (a' — a)  <  (bf — a)  <  (c' — a)  < 

Hence,  in  equation  (3),  the  first  exponent,  a — a',  is  the  least  of  aU> 
But  we  observe, 

1st.  The  exponent,  a' — a,  can  not  be  a  positive  quantity;  for  in 


304  SERIES. 

that  case  the  term  containing  it  would  reduce  to  zero  when  x  =  0, 
(367,  1),  and  we  should  have  A  =  0,  which  is  contrary  to  the 
implied  conditions  of  the  proposition, 

2d.  The  exponent,  a' — a,  can  not  be  a  negative  quantity  ;  for  in 
that  case  the  term  containing  it  would  reduce  to  infinity  when 
x  =  0,  (367,  2),  and  we  should  have  A  —  oc,  which  is  also  con- 
trary to  the  implied  conditions  of  the  proposition.  Now  since  a' — a 
can  neither  be  a  positive  nor  a  negative  quantity,  it  must  be  nothing 
or  zero ;  that  is, 

a' — a  =.  0,     or     af  =  a. 

It  follows  also  that  each  of  the  other  exponents,  b' — a,  c' — a, 
etc.,  in  equation  (3),  is  positive,  being  algebraically  greater  than  zero ; 
hence  all  the  terms  after  the  first  in  the  second  member  of  this 
equation,  must  disappear  when  x  =r  0,  (367, 1),  and  we  shall  have, 

A  =  A'x°  =  A'. 
Now  since  A  and  A'  are  independent  of  x,  we  shall  have 

Axa  —  A'xaf, 

whatever  be  the  value  of  x.  We  may  therefore  suppress  these 
terms  in  equation  (1).  There  will  result 

Bx*+  Cxc-\- ....  =  B'y?'+  CV'+ . . . . , 
whence,  by  reasoning  as  before,  we  shall  find  that 
b  =  //,      c  =  c',     etc. 
B  =  B'     C=  C',   etc. 

That  is,  equation  (1)  is  an  identical  equation,  the  two  members 
having  the  same  form.  Hence,  the  given  equation  is  also  identical, 
and  the  proposition  is  proved. 

It  is  obvious  -that  the  preceding  demonstration  will  apply  if  one 
or  more  of  the  exponents,  a,  l>,  c,  ....  are  negative ;  or  if  a  =  0, 
in  which  case  each  member  will  contain  an  absolute  term. 

III.  In  every  equation  which  is  satisfied  for  any  value  whatever 
of  the  unknown  quantity,  and  which  involves  like  powers  of  this 
quantity  in  the  two  members,  the  coefficients  of  the  corresponding 
powers  will  be  equal,  each  to  each. 
Let  us  assume  the  equation, 

Aar+Bx*+  Cx*-\- ....  =  J  V-f^V-f  <7'V+ 
the  number  of  terms  bein<:  either  limited  or  unlimited. 


IDENTICAL    EQUATIONS. 

Now  if  this  equation  is  capable  of  being  satisfied  for  any  value 
of  x,  then  according  to  the  preceding  demonstration,  not  only  must 
the  exponents  of  x  in  the  two  members  be  equal  respectively,  but 
the  coefficients  also  must  be  equal,  each  to  each  ;  that  is, 

A  =  A',     B  =  B',     0=  C",  ;etc. 

Every  such  equation  is  obviously  identical,  though  it  is  not  neces- 
sary that  A,  B,  C,  etc.  should  be  of  the  same  form  respectively, 
as  A',  B1,  C',  etc. 

IY.  In  every  equation  which  is  satisfied  for  any  value  whatever 
of  the  unknown  quantity,  and  which  has  zero  for  one  of  its  members, 
the  coefficients  of  the  different  powers  of  the  unknown  quantity  are 
separately  equal  to  zero. 
Let 

Ax*+Bx*+  Cxc+Dx*+ . . . .  =  0,  (1) 

represent  the  equation,  arranged  according  to  the  ascending  powers 
of  x.     The  coefficients,  J.,  B,  C,  D,  etc.,  are  supposed  to  be  inde- 
pendent of  x,  and  consequently  the  same  for  all  values  of  x. 
Divide  every  term  in  this  equation  by  xa ;  we  shall  have 

A+ Bx*-a-{-  Cx*-*+Dxd—+ =0.        (2) 

In  this  equation  make  x  =  0 ;  then  since  the  exponents,  b — a, 
c — a,  d — a,  etc.,  are  all  positive,  every  term  after  the  first  will 
reduce  to  zero,  (3G7,  1),  and  we  shall  have 

A  =  0. 

Suppressing  Ax9  in  equation  (1),  and  then  dividing  through  by 
x*,  we  obtain 

B+  Cx*-*+Dx*-*+ . . . .  =  0.  (3) 

In  this  equation  make  x  =  0,  and  we  have 

^=0. 

In  like  manner  we  may  prove  that  each  of  the  other  coefficients  is 
equal  to  zero. 

It  is  important  to  observe  in  this  connection  that  the  coefficients, 
A,  B,  C,  D,  etc.,  must  be  supposed  to  represent  polynomial  exj)res- 
sions,  ichich  reduce  to  zero  in  consequence  of  having  positive  ana 
rrfji+ive  parts  that  are  respectively  equal  to  each  other. 

26*  '  u 


306  SERIES. 

A 

DECOMPOSITION  OF  RATIONAL  FRACTIONS. 

3O9.  By  means  of  the  properties  of  identical  equations,  a 
fraction  may  often  be  separated  into  two  or  more  partial  fractions, 
whose  denominators  shall  be  simpler  than  the  given  denominator. 
In  every  such  case,  the  given  fraction  is  the  sum  of  the  partial 
fractions  ;  hence  its  denominator  will  be  a  common  multiple  of  the 
denominators  of  the  partial  fractions. 

O  q-1 

1.  Separate  —  -  —  into  partial  fractions. 

By  inspection,  we  perceive  that 

x2—  7z+10  =  (x—5)(x—  2). 
Now  assume 

8*-3L         JL_       JB_  ( 

(x—  5)O—  2)  ~x—  5  "t"  x—  2" 

Since  the  first  member  is  simply  the  sum  of  the  two  fractions  in  the 
second  member,  this  is  obviously  an  identical  equation.  Clearing 
of  fractions  and  uniting  terins,  we  have  .  ._« 


Sx—  31  =  A+£)X—  (24+55),  (2) 

in  which  31  in  the  first  member,  and  (24  +55)  in  the  second,  may 
be  considered  as  coefficients  of  x°.  Now  according  to  (3G8,  III), 
the  coefficients  of  the  like  powers  of  x  in  the  two  members  must  be 
equal  ;  and  we  have,  therefore, 

4+5  =  8,  (3) 

24+55=31.  (4) 

From  these  equations  we  readily  obtain 


whence  from  equation  (1),  we  have 

3 


An*- 


X»_7X_|_10  -  x— 5  7  x 

It  should  be  observed  that  equations  (3)  and  (4)  are  the  equations 
of  condition,  which  must  exist  in  order  that  equation  (1)  shall  be 
true  for  all  values  of  x. 


DECOMPOSITION    OF    FRACTIONS.  307 

1x*  I  x 

2.  Separate  -  —  --  —    into  partial  fractions. 
(*+l)(2x—  1) 

Suppose,  if  possible, 

lx*+x  A 


clearing  of  fractions,  we  obtain 

7x*+x  =  (2A+J3)x+(B—  A); 
transposing  all  the  terms  to  the  first  member,  we  have 

—JB    =  0.  (2) 


If  this  equation  be  possible,  it  must  be  an  identical  equation  ; 
and  as  one  member  is  zero,  the  coefficients  of  the  different  powers  of 
x  must  be  separately  equal  to  zero  (368,  IV)  ;  and  we  shall  have 

7  =  0, 

which  is  absurd.     Hence,  we  infer  that  the  fraction  can  not  be  sep- 
arated into  partial  fractions,  having  numerators  independent  o/*x. 
Again,  assume 

7x'-\-x  Ax  Bx 


clearing,  of  fractions  and  collecting  terms, 

7*'+  x  =  (2A  -f  B}x*+(B— 
equating  the  coefficients  of  like  powers  of  x, 
2A+J3  =  7, 
B—  A  =  1  ; 

whence  we  obtain  A  =  2,  j5  =  3  ; 

and  by  substitution  in  equation  (1), 

2x  3x 


Ans- 


From  this  example  we  learn  that  if  we  assume  an  impossible  form 
for  the  partial  fractions,  the  fact  will  be  made  apparent  by  some 
absurdity  in  the  equations  of  condition. 

NOTE. — If  the  given  denominator  consists  of  three  or  more  factors, 
there  will  be  three  or  more  partial  fractions.  But  there  will  always  be  as 
many  equations  of  condition  as  there  are  numerators  to  be  determined. 


308  SERIES. 


EXAMPLES   FOR  PRACTICE. 

7/y 24 

1.  Resolve     a       — — j—  into  partial  fractions. 

X  — —  JJC— j—  L-x. 

5  2 

Ans.  —  -J . 

x— 1         x—2 

2.  Resolve      a  ,  >>       9n  ^nto  partial  fractions. 


-„      ,  Gar2  —  22ar+18 

.  Itcsolve   .         vx-i  —  -  --  rr   into  partial  fractions. 


x  1  2 
4.  Resolve  —  p^--  into  partial  fractions. 

£C  vC 

1 

!- 


5.  Resolve  —  —  Q7r  into  partial  fractions. 

CC  -  ioJC  -J-oO 

-1-     ^_ 

" 


THE  RESIDUAL  FOR3IULA. 

3  TO.  It  lias  been  shown  in  (89,  4)  that  xm  —  ym  is  exactly 
divisible  by  x  —  y,  if  m  is  a  positive  whole  number.  The  form  of 
the  quotient  is  as  follows  : 


the  number  of  terms  in  the  quotient  being  equal  to  m. 

Now  suppose  x=y;  then  each  term  will  become  a1"-1,  and  since 
there  are  ?n.  terms,  we  have  the  formula, 


The  subscript  equation,  y  =  x,  is  used  to  indicate  the  condition  un- 
der which  the  first  member  of  (A]  will  be  equal  to  the  second. 


RESIDUAL    FORMULA.  309 

371.  We  will  now  show  that  this  formula  is  true,  whatever  bo 
the  value  of  m.     There  will  be  two  cases  : 
1st.    When  m  is  positive  and  fractional. 

r  r.       r- 

Assume     m  =  -•  then  XM  —  ym  =  x'  —  y'. 


_ 

Let  x  '  =  z  j    then    x*  =  zr,    and    x  =.  «*. 

J_ 

lso  let  y  •  =  u  }    then    y 

By  proper  substitutions  we  have 


J_  r_ 

Also  let  y  •  =  u  }    then    y*  =  ur,    and    y  =  u*. 


X—  y  sf—u'         Z'—U' 

z  —  u 

Now  suppose  x—y,  then  z  =  u  ;  and  since  r  and  s  are  positive 
whole  numbers,  we  have  from  (1)  j»     - 

" 


v 

u  / 


r    r  i 

.-       r)  z—  u     «—         rz-        r  r    —i 

= 


Hence,  the  formula  is  true  when  the  exponent  is  positive  and  frac- 
tional. 

2d.    Wlien  m  is  negative,  and  either  integral  or  fractional. 
Suppose  the  exponent  of  x  and  y  to  be  — m;  we  shall  have  ^ 


Now  suppose  x  =  y ;  then  whether  m  be  integral  or  fractional, 
we  shall  have,  from  the  principles  already  established, 

hence, 


=(—-")  X  (m.-)  = 


Hence,  the  formula  holds  true  universally. 


310  SERIES. 

BINOMIAL  THEOREM. 

372.  The  Binomial  Theorem  has  for  its  object  the  develop- 
ment of  a  binomial  with  any  exponent,  into  a  series.     This  theorem 
is  expressed  by  an  equation,  called  the  Binomial  Formula. 

373.  It  is  required  to  expand  (a-j-x)n  into  a  series,  n  beiny  any 
real  quantity  ,  positive  or  negative,  entire  or  fractional. 

We  observe  that 

a+x  =  aYl  -|-  -  }  >,    therefore   (a-j-z)11  =  a*  (  1  -f  -\  . 

(x  \n 
1  _j  —  J  ,  and  then  multiply  the  result 

by  an,  we  shall  have  the  expansion  of  (a-\-x)n. 

Put  z  =  -  •  then  (l  +  -  Y  =  (1+  *)". 

,    |     %     Let  us  UQW  assume  the  equation, 

(!+*)»  =  A+Bz+  Cz*+Dz*+Ez*+  ....  (I) 

in  which  A,  B,  C,  D,  etc.,  are  independent  of  z.  "We  are  to  find 
the  values  of  these  coefficients  which  will  render  equation  (1)  true 
for  all  possible  values  of  z. 

Suppose  z  =  0  ;  then  from  equation  (1),  we  have  A  =  1. 
Hence,  the  assumed  development  becomes 

(l+z)w=  1-hflH-  Cfef+0*i+J5k4+....  (2) 

for^ll-  values  of  z.     Put  z  =  u  .  ;  then 

.    \+.(\+uy  =  l+Bu+Cu'l+Du*+Eu'+....  (3) 

„      -  Subtracting  (3)  from  (2),  and  dividing  the  result  by  z  —  «,  we  obtain 

Vi+.y.-(i±«)!=         /.^x       ^N  (4) 

2  -  U  \    Z  -  U    I  \    Z  -  U  I 

Let  P  =  1-j-z,  and  Q  =  1  -f?*  ;   then   P—  Q  =  2  —  w. 

Equation  (4)  now  becomes 

'  ....     (5) 


Now  suppose  z  =  u  ;  then  P  —  Q.     And  by  the  Residual  Forn> 
ula  (37O),  we  shall  have  i 


BINOMIAL    THEOREM. 


311 


)    =*>•> 

«/•— 


z'—u*  \ 

-  )          =  42*,  etc. 
z—  u  /«=* 

Substituting  these  values  in  equation  (5),  we  have 


Multiplying  both  members  of  equation  (6)  by  (l-j-2),  gives 


-f 


-f-26r 


Multiplying  both  members  of  equation  (2)  by  n,  gives 


CO 


____  (8) 

Now  by  equating  the  second  members  of  (7)  and  (8)  we  shall  have 
an  identical  equation,  because  it  may  be  satisfied  for  any  value  of  z. 
Therefore  the  coefficients  of  the  like  powers  of  z  in  equations  (7) 
and  (8)  are  equal,  each  to  each  (3O8,  III),  and  we  shall  have 


=n£,    or    C  =  B  -, 


ZD+2C=nC,    or    2>= 


=  wD,    or   E  = 
Therefore,  the  values  of  the  coefficients  arc 


;  etc. 


n(n—  l)(n—  2) 
23* 


312  SERIES. 

Substituting  these  values  in  (1)  we  have 


o 


and  by  restoring  the  value  of  z,  which  is  - , 


or.  finally,  multiplying  both  members  of  (&)  by  an, 


Equation  (c)  is  the  binomial  formula,  as  it  is  usually  -written.  It 
will  be  observed,  however,  that  in  the  three  equations,  («),  (*),  (c\  the 
coefficients,  or  the  factors  depending  on  n,  are  the  same  ]  and  in 
practice,  either  (#),  (6),  or  (<$  may  be  employed,  according  to  the  form 
of  the  binomial  to  be  expanded. 

31  74:.  By  inspecting  the  general  formula  (c\  we  perceive  that  in 
the  expansion  of  a  binomial  in  the  form  of  a-\-x,  the  law  of  the 
exponents  is  as  follows  : 

1.  —  The  exponents  of  the  leading  letter  in  flie  successive  terms 
form  a  scries,  commencing  in  the  first  term  with  the  exponent  of  the 
binomial,  and  diminishing  by  1  to  the  right. 

2.  —  The  exponents  of  the  second  letter  form  a  scries,  commencing 
in  the  second  term  with  unity,  and  increasing  by  1  to  the  right. 

And  the  law  of  the  coefficients  is  as  follows  : 

3.  —  The   coefficient  of  the  first  term  is  unity,   and  that  of  the 
second  term  is  the  exponent  of  the  required  power. 

4.  —  If  the  coefficient  of  crny  term  be  multiplied  by  the  exponent 
of  the  leading  letter  in  that  term  ;  and  divided  by  the  exponent  of 
the  second  letter  plus  1,  the  remit  will  be  the  coefficient  of  the  fol- 
lowing term. 

37«>.  If  we  take  the  least  factor  in  each  of  the  successive  coef- 
ficients of  the  expansion,  commencing  at  the  second,  we  have  a  de- 
creasing series 

»,     («—]),     (Ti—2),     (n—  3),     etc., 
in  which  the  common  difference  is  unity. 


BINOMIAL    THEOREM.  313 

Suppose  n  to  be  a  positive  integer,  then  the  least  factor  in  the  nu- 
merator in  the  (w+2)d  term  will  be  (H  —  »),  or  0,  and  this  term 
will  disappear.  But  if  n  is  negative  or  fractional,  then  no  one  of 
the  factors,  (M  —  1),  (»—  2),  (M  —  3),  etc.,  can  be  zero,  and  the  expan- 
sion may  be  continued  indefinitely.  Hence,  ^ 

1.  —  When  n  is  a  positice  intfyer,  the  expansion  of  the  binomial 
will  be  a  finite  series,  the  number  of  terms  being  n-j-1. 

2.  —  When  n  is  negative  or  fractional,  the  expansion  of  the  bino- 
mial will  be  an-  -infinite  series. 

APPLICATION   OF   THE   BINOMIAL  FORMULA. 
370.  Let  us  resume  the  equation, 

7i(».  —  1)  7/(M—  !)(»—  2) 

(a+x)*  =  a*+na»-*x+  --  -4y—  V~V  -f-  -    0    .    8  —  an~V  +  ....   (e) 

If  n  be  entire  and  positive,  this  formula  will  be  an  expression  of 
involution,  denoting  some  power  of  the  binomial. 

If  n  be  fractional  and  positive,  the  formula  will  be  an  expression 
of  evolution,  denoting  some  root  of  the  binomial. 

If  n  be  negative,  the  formula  will  express  the  reciprocal  of  some 
power  or  root  of  the  binomial. 

377.  Since  the  binomial  coefficients  depend  entirely  upon  the 
exponent  ?*,  they  may  be  formed  independently.  To  do  this,  we 

have  simply  to  commence  with  unity  and  multiply  by  n]          >  —  TT~} 

—  o 

etc.,  continually. 

1.  Expand  (a  —  j-)6  into  a  scries. 

Here  n  =  G  j  hence, 

The  first  coefficient  is  1=1 

"    second  "           "  lyC  =     G 

«    third      "           "  GX:]  =  15 

"    fourth    "           «  16X3  =  20 

«    fifth       "           «  20  XJ  =  15 

"    sixth      "           "  15  X?   =     G 

"    seventh"           "  GX      =     1 


Since  the  odd   powers  of  —  x  arc  negative,  we   have  for  the  literal 
factors  of  the  terais, 


SERIES. 


ffl«,  —a'x, 


Therefore  the  expansion  will  he 

(a—  a;)'  =  a6—  6a^+15a.V—  20«V-f  15aV—  C^'-f  x«. 
2.  Expand  (a-j-a:)-  into  a  series. 

In  this  example  n  =  i.     Keprcsent  the  coefficients  by  A,  J5,   C, 
D  ----  ;  then 

A=  +  1 

Ji  =  AX       n       =+ 


The  literal  factors  of  the  terms  will  be 
a?,    a~vx,    <T*x*,    a~2x8,     , 
Hence,    (a-f-^O^  = 


24  2-4-6  2-4-G-8 

or  by  taking  out  the  factor  a2,  in  the  second  member, 


1        -a  3  8'5         _4  \ 

2tt"       "24 a"         h24:6at:c   ~24¥8a"    f*V>:vJ 
or  by  clearing  of  negative  exponents, 


We  might  have  obtained  this  last  result  directly,  by  putting  the 

binomial  in  the  form  of  a~  (  1-| J  5.     It  is  well,  however,  to  note 

the  transformations  made  above. 


BINOMIAL   THEOREM.  315 


3.  Expand  a  into  a  series. 

Observe  that 


Whence,  by  expanding  the  factor  f  1  -j —  )     we  obtain 


4.  Expand  (a'— x*)*  into  a  series. 

If  we  take  the  descending  powers  of  a*,  commencing  with  the  . 
5th,  and  the  ascending  powers  of  xa,  commencing  with  the  first,  we 
have  for  the  literal  factors  of  the  terms, 

16         n^nr*         n*-r*         n*Y*         /r8?-8         fl* 
CL     ,        Cp    £C  i        tt.t/j        Ui  JC  ,        U  JL   j        •C-.- 

Hence,  with  the  coeflicients  the  development  becomes 
V-flOaV — 10aV-j-5aV- 


EXAMPLES  FOB  PRACTICE. 

.  •  •  «. 

1.  Find  the  fifth  power  of  a—  b.  *  t 

Ans.  a'—  5a«6+10a«i7—  lOa'^ 

2.  Find  the  sixth  power  of  1+c.    A  ^cW^/V 

Ans.  !-|-6c4-15ca4-20c8-fl5c4-f6c'-fca. 

3.  Find  the  seventh  power  of  x-\-y. 

Ans.  x7+7^V-h-n^y+35xy+35xy-f21xy+7xy+y'. 

4.  Find  the  eighth  power  of  a3  —  1. 

Ans.  a16—  8au+28a"—  56a10-j-70a»—  56a'+28a4—  8aa+l. 

5.  Find  the  ninth  power  of  a  —  c. 

.  a9—  J)a8c  -j-  36a'c2-r84aV  -f  126a5c4—  126aV  +  84aV— 


6.  Expand 
\ 

7.  Expand  (aa—  a:')6. 


316  SEBIES. 

8.  Expand  (*«—  z4)'. 

Ans.  z10—  SxV+KW—  10xVf+5:cV- 

9.  Expand  (o'^-f  cfy9)6. 

.  a1  V-f  6a" 


—  a:3  i 


^j  10.  Expand  (a  —  a:)3  into  a  scries.    C^-^JL  *^wt   */  #t  #, 

--h-i^-i     .        /    /Y_^  x_  __  x9          .  3x*  -  3-5a* 

i—**-rf  V    ""^a      2-4a8~"2-4-6^"~"2-4-G-8a4""" 

ir'^'r1  i 

11.  Expand  (1—  x)*  into  a  series.  ^ 


3       3-6      3-6-9       3-6-9-12 


f.  , 

^^.Js^fl2.  Expand  (a+1)*  into  a  series. 


13.  Expand  (a-J-i)3  into  a  series. 


J/!   ,  1      jg_   ,  .2-Sy         2-6-Sf  X 

\     ^3a      3-Oa'  ^  3-G-Oa1      3-G-9-l:V^      "/ 

-f    14.  Expand  -L 


,  a        -4t- 

LX  ^4^  y.i1^-,  Expand      _      1nt6  a  scri 

XTr1"«f)<-T*-4      <^ix 

<       /  J/A    16.  Expand 
/ 


a--^a2  into  a  series. 


_ 
~~ 


, 

,  2^1  l1'  Expand  (a—  c2)^  into  a  series.  ^_^  ^  ^  ^.. 

'"         f  /.       2ca         2c4  2-4ca        (J2  '  4  •  7ca  \ 

jSSi  a   \    '  ~  3^  ~  3T6^~a  "  '  3  •  6  •  9  a  3  ~~  3  •  6  •  9  •  1  2  a4  "       "J 

18.  Expand  ^c'-j-  xa)~2  into  a  series.  ^  </<?  ^  ~ 


>**    An,          l--A    -  ' 

/       ?    ,    '     j   c\        2c'"1"2-4c4       2-4- 


6c8 


BINOMIAL    THEOREM.^./     tf^-J 
£^J*A=      £ 

19.  Expand  (1  —  a)~3  into  a  series.      -  _^  t,*-^  ~     -JQ 
Am.  ; 


^20.  Expand^  (a3  —  x3)*  into  a  series.   ^-/  ^-^r  <  ^V^V'^-'^ry 

"TT^  *  f*f&*~  3x9        3x4  3-5x'  3^-9x*        C^"**-  A. 

Am.  ya  [a  —  —  -  ^  -  ^j^§  -  4.8.'1:2.16ttf  —  •  •  •  •  j 

,  ,^   21.  Expand  (a+y)~4  into  a  series.  <»r^  /i*.*y  +  tc  *r6y*-^  .  - 
f*/^  1        %       ICy        2(y       35;/        56^ 

f.-^  '    a«~VH       a6    '    "    aT          ~~^~~      '  ~tf~ 

^  22.  Expand  ^=  into  a  scries.  *  ^  V=  W/-/  ^ 

AA*.,       6r,        6.lb.4      G.n.16r. 
ns'  r+  5  +^  +  ^3^3 


f.  Expand   ^1 — x*  into  a  scries.     -^C/TXV 
-a:4       14x*       14-29.*13       14-2944x1§ 


^    METHOD   OP 

378.  In  the  formula  (x+y)n  = 


we  may  suppose  cc  and  y  to  represent  any  quantities  whatever  ;  and 
thus  we  may  obtain  the  development  of  the  powers  of  binomiala 
with  numerical  coefficients,  or  of  polynomials. 

1.  Involve  3a-J-2c  to  the  fifth  power. 

The  binomial  coefficients  for  the  fifth  power  are 

1,     5,     10,     10,     5,     1. 

And  by  connecting  these  with  the  powers  of  the  given  terms, 
according  to  the  law  of  the  formula,  we  have 

(3./+2c)5  ==  (3a)6-f  5(3a)4(26-)-hlO(3a)8(2c)2-i-10(3a)'(2c)s-{- 
5(3a)(2c)<-K2c)'; 

or,  by  performing  the  operations  indicated, 
(3a-f  2c)6  =  243a'-f  810a4c-fl080aV+720aV-f  240ac4-f  32c* 


318  SERIES. 

2.  Involve  a-f&-f2c*  to  the  fourth  power. 

"We  may  consider  the  polynomial  in  two  parts,  a-f&,  represented 
by  cr,  and  -|-2ca  represented  by  y.     Then  we  have 

(a.  4-  i  +  2c')«  =  (a  +  i)*+  4(a  +  i)'(2c')  +  6(a  -f  &)f(2c')«  + 


Performing  the  operations  indicated, 

2t-2)*  =  a4+  4aH6i(-  6a262-f  4a&s-f  6«-f  8aV+  24a' 


EXAMPLES  FOR   PRACTICE. 
\ 

1.  Find  the  third  power  of  a  —  2b. 

Ans.  a9—  6aaZ>-{-12a&a—  SI9. 

2.  Find  the  fourth  power  of  2a+3x. 

AM.  16a4+96a'a:+  216a^a 

3.  Find  the  fourth  power  of  1—  \a.  -/-4{£j 

Ans.  1—  Sa+fa'—  ^a' 

4.  Find  the  fourth  power  of  a*  —  ax-fa*. 

Ans.  «8-4a7.i-  +  lOaV  —  ICaV  +  19aV  —  L6aV  -f-  lOoV—  " 


5.  Expand  (4aa—  3x)?  into  a  scri 

-Ji.-        ..     .     _      \ 

16a9        512a4        24576a'          "V 


FRENCH'S  THEOREM. 

3  TO.  When  a  binomial  having  numerical  coefficients  is  to  bo 
raised  to  any  power,  the  coefficients  of  the  expansion  may  be  ob- 
tained with  great  facility  by  means  of  a  simple  modification  of  the 
binomial  formula.  We  have  (z-f-*0w  = 

n(n—V)  n(n—  IVn—  2) 

s«+n*-'iH-  —  ir—  •  a^VH-        >,     A   o       ^3^+  -  •  •  • 

6  A  i) 

In  this  equation  make   z  =  ax,  and  u=.ly}  then 


BINOMIAL    THEOREM.  319 

in  which  a  and  b  may  represent  the  numerical  coefficients  of  x  and 
y.     Now  denote  the  numerical  coefficients  of  the  expansion  by  C19 


C2)  C$,  etc.     We  shall  then  have 


in  which  Cl     = 


c          c-   n    -b 

C*  °*       I         a' 

r          r  •  "-1  -  b 

°*  °*     ~2~     a' 

n-2b 


c 

64 

(7 
°5 


4        a 

1.  Find  the  fourth  power  of  ba-\-ox. 
In  this  example  we  have 

n  =  4,     a  =  5,     6  =  3, 

and  the  coefficients  of  the  expansion  will  be 

C  ,  =        5*  =625 

C2  =  625  -  f  •  |  =  1500 
C  3  =  1500  -f  •  |  =  1350 
<74  =  1350  •  f  •  |=  540 
C5  =  540  •  i  -  |=  81  Hcnco, 

(5a-f  3x)4  =  625a4-fl500a»z+1350aV+540az'-f81.r4,  Ans. 

2.  Find  the  fourth  power  of  — 

2  4          .6436 

We  have  n  =  4,  a  =  —  ,    6  =  —  ,    and    -  =  —•  —  =  -• 
o  5  a       5    2       5 

Hence  the  coefficients  of  the  expansion  are 

c,  (!)'  =  if 

c-z=  if-f-f  =  t3i 
t-Y3  =  ill  '  1  '  f  =  W 

<^  =  w  •  i  •  f  =  m 

<?•  =  m  •  i  •  i  =  nt  ncn°«. 

2c       4a=\4_^6   4     128  128  ,  ,      512  256    4 

T"  5      =-  81C  ~1MCX"  75  ca:    ~375ca!    *~  625* 


320  SERIES. 


EXAMPLES    FOR    PRACTICE. 

1.  Find  the  fourth  power  of  2.r-foy. 

Aug.  lG.i;4-j- !GO.rV-}-00(Uy -f  1000.ryf-f  G25/. 

2.  Find  the  fifth  power  of  2a— 3.r. 

An*.  32«b— 240</'.c-f720«V— lOSOaV-j-SlOa*'— 243*'. 
o.  Find  the  sixth  power  of  o-j-4.r'. 

vb*.  729+5832-c1 +10-140^+  345GOa:-+3i5GOjci+  lS4S2xlf+ 
4Q9G-C11,. 

4.  Find  the  fourth  power  of  --  -f.  —  . 

•A          o 

Ans.  &sa'+$la'r+ZiaV+\$laS+  jSjr'. 

2£        or 

5.  Find  the  sixth  power  of  ~   -|-  —  . 

A>u.  &Y+  5?'V+aa0^f-f20/V>4-J.s^/-4+i^'r*+  W'1'- 

G.  Find  the  fifth  power  of . 

4        5 

m*          7U1         7H8         7?r         m  1 

/I   »JO          I _„   I  _ 

'1024      250  ^  1GO       200^600       3125' 

7.  Find  the  eighth  power  of . 

2        '2m 

m*        mc       7wi4      7??^a        85  7  7 

yl  j)  o      I  __„    I      _____ 

*  250       32  J  04        32   '*"  128  ~~  32m1  "»"  Gi/u* "" 
1  1 

S2w?   '   250m6' 


DEVELOP^IEXT  OF  SURD  ROOTS  IXTO  SERIES. 

SI8O.  The  approximate  value  of  a  surd  root  mny  lie  oLtaincd 
with  much  facility  by  expanding  the  root  into  a  series. 

Let  un  represent  that  perfect  ?>th  power,  which  is  next  less  or 
next  greater  than  the  given  number,  and  let  I  denote  the  difference 
between  this  power  and  the  ijivcn  number.  Then 

an+b,     or     an— b, 
will  express  the  given  number.     But  we  have 


DEVELOPMENT  OF  SUED  BOOTS.          321 


a" 

Developing  the  radical  parts  into  series,  we  have 


—  2 


n   rf  2;i      a3*  "    n     2»        3/i       a8" 

The  second  members  of  these  equations  contain  no  radicals  ;  hence, 
Any    Kurd   may   l>e  developed  into  a  series  of  rational  terms; 
whence  ly   summing  the  series,  we  may  obtain  approximately  the 
indicated  root. 

It  should  be  observed  that  the  smaller  the  fraction  —n  is,  the  more 

rapidly  will  the  series  converge. 

1.  Find  the  cube  root  of  76,  to  six  decimal  places. 
The  smallest  fraction  will  result  by  taking  the  cube  which  is  next 
less  than  76,  or  64  ;  thus, 


We  may  now  develop  the  radical  part  by  equation  (1),  in  which 

q  A  ^    _ 

To  form  the  binomial  coefficients  we  have  the  factors, 

1    _1 

n  ~~3 

l—n  1  1— 4»  11 


2n  3  bn  15 

1— 2?i_  5  1 — 5?i_  7 

3»  ~~9  Qn  ~~jf 

1— 3n  2  1— 6n  17 


4n  3  7??-  21 

We  represent  the  successive  terms  by  A,  B,  C,  etc.;  and  to  secure 
accuracy  in  the  final  result  to  the  6th  place  of  decimals,  wo  should 

V 


322 


SERIES. 


carry  the  computation  in  each  term  to  the  7th  place.     Thus  we  have 


A 

*=  + 

C  =  — 

#  =  — 

^7  =  — 

.F7  =  — 

£  =  — 

//=  — 

Algebraic  sum, 


•A 

-T3S 

•A 


T3* 


c 

D 
E 
F 
G 


1.0000000 
.  .0625000 

.0039062 

.0004069 

.0000508 

.0000069 

.0000010 

.0000001 

1.0589559  =  ^I+A 
4 


4.235824±,  Am. 


Whence, 

2.  Find  the  5th  root  of  25,  to  6  decimal  places. 

The  most  convenient  fraction  will  result  by  taking  that  5th  power 
which  is  next  greater  then  25,  or  32  •  thus, 


Equation  (2)  will  now  apply ;  and  the  operation  will  be  as  follows : 

L_ L 
n~5~ 


1-n 

2 

l—4n            19 

2n   ' 

5 

5?i              25 

l—2n 

3 

1—  5n           4 

3n 

5 

6/1              5 

1—371 

7 

1  —  Qn            29 

4n 

10 

-    77i              35 

A 

=     +  1.0000000 

B  =  — 

J 

'  & 

=     _      437500 

(7  =  — 

1 

'  A  '  & 

=                38281 

D  =  — 

1 

'  A  *  ° 

=     —          5024 

E  —  — 

T77T 

•£>'  D 

=     —            769 

IT     

4S 

•  A  '  E 

=     —            128 

G  =  — 

1 

'•h'  F 

=     —              22 

A 


G    =     — 


Algebraic  sum, 
Whence, 


.9518272  = 
2 


=        1.903654-f, 


EXPANSION    OF    FRACTIONS.  323 


EXAMPLES     FOR    PRACTICE. 

Find  tlio  values  of  the  following  indicated  roots,  to  the   6th 
lecimal  place  : 

1.  V9.  *  Ans.  2.080084. 

2.  VsT  Ans.  3.141381. 

3.  VETO.  Ans.  4.641589. 

4.  VTTO.  Ans.  4.791420. 

5.  V597.  Ans.  3.122851. 

6.  v'oa  Ans.  1.978602. 

7.  1/4.  Ans.  1.319508. 

8.  ^8275.  Ans.  5.047104. 

9.  VT25.  Ans.  1.993235. 

EXPANSION  OF  FRACTIONS  INTO  SERIES. 

381.  An  irreducible  fraction  may  always  be  converted  into  a 
series,  by  dividing  the  numerator  by  the  denominator. 

1.  Convert into  a  series. 

1+a 

Observe  that 


1-f  a  ~  a-fl 
Hence,  there  may  be  two  ways  of  dividing  j 

1st.  2d. 


C-i+i 


a 

1       1 


— a— a' 


824  SERIES. 

The  law  of  expansion  is  obvious  in  both  quotients,  and  we  havo 
from  the  same  fraction  two  series;  thus, 

Y^a  =  1—  «  -f  a2  —  a3  -}-  a4  —  </a  -f  ____       (!) 

—  -  l—  -    I  _1  i_  L    1 

«-f  i  "~  a      «*  +  a3  ~~a4  ~*~  <7»  ~~<?~*~  ---- 

We  observe  that  each  of  the  scries  obtained  is  by  its  law  of  devel- 
opment, an  infinite  series. 


EXAMPLES    TOR    PRACTICE. 

1.  Convert  —  —  into  an  infinite  scries. 

4        T      x 
Ans.  1_-  +-  __.... 

2.  Convert  -   into  an  infinite  scries. 

a  —  x 


8.  Convert  T—  -   into  an  infinite  series. 
1  —  ^ 

Ans.  l+2z-f- 

4.  Convert  —  —  —  ,  into  an  infinite  scries. 
'2 


5.  Convert  -  -  •  —   into  an  infinite  scries. 
1  —  a-}-  a 

.  1-f-a—  as—  a4-f-ae-f-a?—  a'—  a"-f.... 


C.  Convert  - — — -  into  an  infinite  scries. 

I — 2.x — ox 

Ans. 

l-f2x 
7.  Convert — — -t  into  an  infinite  scries. 

X.  •"—^•'•-•-  3^ 

Ans.  1 


INDETERMINATE    COEFFICIENTS.  325 


METHOD  OF  INDETERMINATE  COEFFICIENTS. 


\t  It  is  evident  that  if  a  fraction  be  developed  into  a  series, 
the  equation  which  results  by  placing  the  fraction  equal  to  its 
development  is  capable  of  being  satisfied  for  any  value  of  the  un- 
known quantity;  in  other  words,  it  is  an  identical  equation. 

On  this  fact  depends  an  important  method  of  expanding  an  alge- 
braic expression  into  a  series,  called  the  Method  of  Indeterminate 
Coefficients.  It  consists  in  assuming  the  required  development  in 
the  form  of  a  series  with  unknown  coefficients,  and  afterward  deter- 
ming  the  values  of  the  coefficients  by  means  of  the  known  properties 
of  identical  equations. 

1  -1. 2  c 

1.  Develop  — l-^-  into  an  infinite  series. 
1 — ox 

Assume 

1+2ji==.4|j 
1 — ox 

Clearing  this  equation  of  fractious,  and  transposing  all  the  terms 
to  the  second  member,  we  have 


=  (4—  1)+ 


—SA 


+  D 
— SO 


(2) 


The  term  A — 1  may  here  be  considered  as  the  coefficient  of  x* 
understood. 

Now  because  equation  (2)  is  an  identical  equation,  the  coefficients 
of  the  different  powers  of  x  are  separately  equal  to  zero,  (3CS,  I V\ 
Thus, 

A — 1     =  0,    whence   A  =      1  • 
£—3.1—2     =0,        «        J5=      5; 
C—  3^  =  0,        «         0=    15; 
D— 3(7=0,        "        D=    45; 
E—SD  =  0,        «        E  =  135,  etc. 
Substituting  these  values  in  (1)  we  have 

l4-2x 

—-^  =  l+5*-f-152;'-H5.s'-f  135x4-f. . . . 

28 


326 
2.  Develop 


SERIES. 


a;  —  2x  +6x* 
th 
Therefore,  assume 


an  infinite  series. 


We  perceive  that  the  first  term  of   the  series  must  be  —  or  x~l 

x 


Clearing  of  fractions  and  transposing,  we  have 


0=     A 
— 1 


— 2A 
—1 


(7 


+QA 
Putting  the  coefficients  equal  to  zero, 

A — 1     =  0,    whence,  A  = 
£—2  A— I     =•= 
C—2£+QA=Q,        «         C  = 


=  0 


—2D 

+6(7 


—2E 

+6Z> 


=          3 


0; 


E=—  36; 

F=+  36; 

'=0,         "         G  =  +288,  etc. 
Substituting  these  values  in   the   assumed   development  (1),   and 
observing  that  the  term  containing  C  will  disappear  because  (7=0, 
we  have 


,  =  -  +3—18x2 

— X  -+OX  X 

KOTE. — It  is  not  necessary  to  transpose  the  terms  to  one  member ;  for 
if  neither  member  is  zero  we  have  simply  to  equate  the  coefficients  of 
the  like  powers  of  x  in  the  two  members,  according  to  the  third  property 
of  identical  equations. 

The  method  of  Indeterminate  Coefficients  is  applicable  to  a  great 
variety  of  examples,  but  always  with  this  provision,  viz.  :  That  we 
determine  by  inspection  what  power  of  the  variable  will  be  contained 
in  the  first  term  of  the  expansion,  and  make  the  first  term  of  the  as- 
sumed development  correspond  to  the  knoicn  fact. 

If  the  assumed  development  commence  with  a  power  of  the 
variable  higher  than  it  should,  the  fact  will  be  indicated  by  an 
absurdity  in  one  of  the  resulting  equations.  If,  however,  the  assumed 
development  commence  with  a  power  of  the  variable  lower  than  is 


INDETERMINATE    COEFFICIENTS.  327 

necessary,  no  absurdity  will  arise  ;  but  the  redundant  terms  will 
disappear  by  reason  of  the  coefficients  reducing  to  zero. 


EXAMPLES  FOE  PRACTICE. 


_ 

1.  Develop  -  —    -  into  a  scries. 
1  —  ox 

Ans.  l 


2.  Develop  -  -  -  —  -  into  a  series. 

r  1  —  x  —  x3 

Ans. 

3.  Develop  -  —  -  -  —  -  into  a  scries. 

1  —  ox  —  Lx 

Ans.  l-J-2z-|-8z>-|-28xa+100z4-f-356:e*+  .... 


.r)  . 
4.  Develop  —  —  5—  r-t  into  a  series. 

Ans. 


2 

5.  Develop  -^  -  ^-j-  into  a  series. 
ox  —  £x 


2        4       8x       16x«       32x' 
AnS'  ^  +  9  +  2T  +  -gT  +  243"  +" 


6.  Develop  .  ,  0  ,  ,  0   .  into  a  series. 

v  34 


nto  a  sees. 
—  2a)x—  (2a—  3a'X-|-(3a»—  4a4)x»— 


8.  Develop  \f\  —  x  into  a  series. 


x          x*  ox* 

~  "  ~~          "~ 


2  ~~  24  "~          G  :    2-4-6-S      •••• 

NOTE.  —  Assume  Vl—  x  =  A+Bx+  Ox>+Dz*+  ----  ;  then  square  both 
members,  and  the  equations  for  the  coefficients  will  be  readily  obtained. 


328 


SERIKS. 


9.  Develop  l/l-f-3x-|-5x8-f  7x3-j-9x4-|- into   a   series  of 

rational  terms. 

«-       11*'       23*'       179x*_ 

I     i  •  •  •  • 


10.  Develop 
alent  scries. 


+ 


*'+  x4 
Ans. 


mtoancquiv- 


REVERSION   OF   SERIES. 


383.  The  Reversion  of  a  Series  is  the  process  of  finding  the 
value  of  the  unknown  quantity  in  the  scries,  expressed  in  terms  of 
another  unknown  quantity. 

1.  Given  y  —  ax-{-lx'2-\-cx*-\-dx*-}-cx*-\- ,  to  find  the  valuo 

of  x  in  terms  containing  the  ascending  powers  of  y. 

In  this  equation,  x  and  y  arc  two  indeterminate  quantities,  and 
either  may  have  any  value  whatever  without  altering  the  form  of 
the  series.  We  may  therefore  apply  the  method  of  Indeterminate 
Coefficients.  Assume 

We  may  now  find  by  involution  the  values  of  »*,  ccs,  x*,  x*,  etc., 
carrying  each  result  only  to  the  term  containing  y*.  Then  substitut- 
ing for  T,  xa,  x3,  etc.,  in  the  given  equations  we  shall  have,  after 
transposing  y, 

0  —  oA  I  y-\-  aB 

—II         I  A1        4-21AB        +21AC 

+      IB9 


+      cA 


+     dA* 


-f 


This  is  an  identical  equation,  being  true  for  all  values  of  y. 
And  if  we  place  the  coefficients  of  the  different  powers  of  y  sepa- 
rately equal  to  zero,  (368.  IV),  and  reduce  the  resulting 


REVERSION   OP   SERIES. 


329 


tions,  wo  shall  obtain  the  values  of  the  assumed  coefficients  as 
follows : 

A=    L 

a 


If  wo  substitute  these  values  of  A,  E,  (7,  etc.,  in  equation  (t),  wo 
shall  have  the  value  of  x  in  terms  of  a,  2»,  <•....,  and  the  powers 
of  y  ;  that  is,  the  given  series  will  be  reversed. 

2.  Given  y  =  a.x-\-lx*-\-cx*  -\-djc1     e 
of  x  in  terms  of  y. 

Assume         x  =  Ay+fy9+ 

Proceeding  as  before,  we  shall  obtain, 


,  to  find  the  valuo 
..         (1) 


In  the  preceding  examples  the  letters  a,  I,  c, . . . .,  represent  any 
coefficients  whatever.  Hence,  in  reverting  any  series  in  cither  of 
these  forms,  we  may  determine  the  values  of  the  assumed  coefficients 
by  an  application  of  formula  (#"),  or  (G). 

3.  Revert  the  series  y  —  x-4-2x2-}-4x3-f-8x4+ 

Assume,        x  =  Ay+By*+  C/+  Dy'+ .... 
28* 


330  SERIES. 

If  we  now  substitute  in  formula  (F)} 

a  =  1,  b  =  2,  c  =  4,  d  =  8, 

we  shall  obtain        ^1  =  1,  B  =  —  2,   (7  =4,  D  =  —8. 
Hence,  x  =  y—  2yf  +  4y8—  8^4-|-  .  .  .  .  ,  ^n«. 

EXAMPLES    FOR   PRACTICE. 

1.  Revert  the  series  y  =  x+x*-\-  x*-±-x4-\-  x  s+  .... 

Am.    X  =  y—y*+y*—y*+y* 

2.  Revert  the  series  y  =  z+Sx'-f  5x*-f7z4-f  9x'-f  .  .  .  . 

-An*.  *= 


3.  Revert  the  series  x  =  y  —  ^_L^  _  ?.  _L  ?.  _ 

2  ^3        4  +5 

S^=x+i~2  +  r^3  +  rl^4  +  r^ 

4.  Revert  the  scries  y  =x—  x'+x*  —  x7-f-x9—  xn-|-  .... 


5.  Revert  the  series  y  =  2x-j-3x8-j-4x6-f-5^T-f  ____ 


6.  Revert  the  scries  x  =  2^+4yf-f6y8+8/-f-10yft-f- 


-^-        _____.... 

384.  One  of  the  principal  objects  in  reverting  a  series  is,  to 
obtain  the  approximate  value  of  the  unknown  quantity  when  the 
sum  of  the  series  is  known.  Thus, 

1  .  Given  ^  =  2x  --  3~  +  ~^  ---  f~+  ____  to  find  the  aP~ 
proximate  value  of  x. 

4z*       6^8       8x4 
Letusput        s  =  2x—  —  +  —  --  —  -j-  ----  ,        (i) 

and  consider  x  and  s  as  variables.     Reverting  (l)y  by  formula  (^), 


8  S«          1388 

=     ---- 


Now  if  we  put  s  =  £  in  this  equation,  the  result  will  bo  a  con 


REVERSION   OF   SERIES.  331 

verging  series ;  and   we  may  find  the  approximate  value  of  x,  by 
computing  the  values  of  the  terms  separately.     Thus, 


21    =         '         ='125000 


1512  ~  256 ' 1512 


,.000013 


Hence,  x  =  .135993, 


EXAMPLES. 


1.  Given  |  =  5x—  20^'-f80xs—  320x4+1280x6—  ....,to  find 
the  approximate  value  of  x.  Ans.  x  =  .117647. 


=*+++++  .....  tofind 

the  approximate  value  of  x.  Am.  .454620. 

1  x*       x*         x1 

3.  Given  -  =   x  —  Q  —  |Q  —  jp>  f-  ----  >  to  find  the  approxi 

mate  value  of  x.  Ans.  x  =  .201369. 

4.  Given  -=  *—  ^  +  ^-^g  +••-•>  to  find  thc  aP" 
proximate  value  of  x.  Ans.  x  =  .274655. 


SUMMATION  OF  INFINITE  SERIES. 

385.  The  Summation  of  a  Series  is  the  process  of  obtaining  a 
finite  expression  equivalent  to  the  series. 

38G.  The  method  of  summing  a  given  series  must  always 
depend  upon  the  nature  of  the  series,  or  the  law  governing  its 
development.  Formulas  have  already  been  given  for  the  summa- 
tion of  arithmetical  and  geometrical  series.  We  will  now  investigate 
the  methods  of  summing  a  variety  of  other  series. 


332  SERIES. 

RECURRING  SERIES. 

387.  A  Recurring  Series  is  one  in  which  a  certain  number  of 
consecutive  terms,  taken  in  any  part  of  the  series,  sustain  a  fixed 
relation  to  the  term  which  immediately  succeeds.  Thus, 


is  a  recurring  series,  in  which  if  any  two  consecutive  terms  be  taken, 
the  product  of  the  first  by  3x*  plus  the  product  of  the  second  by 
2.r,  will  be  equal  to  the  next  succeeding  term.  The  coefficients  of 
these  multipliers,  or 

3,    2, 
arc  called  the  scale  of  relation.     In  the  recurring  series 


the  scale  of  relation  is  8,     2,     1. 

388.  A  recurring  series  is  said  to  be  of  the  first  order,  when 
each  term  after  the  first  depends  upon  the  term  which  immediately 
precedes  it.     The  scale  of  relation  will  consist  of  a  single  part,  and 
the  series  will  be  a  geometrical  progression. 

A  recurring  series  is  said  to  be  of  the  second  order,  when  each 
term  after  the  second  depends  upon  the  two  preceding  terms;  the 
scale  of  relation  consists  of  two  parts. 

A  recurring  scries  is  said  to  be  of  the  third  order,  when  each 
term  after  the  third  depends  upon  the  three  preceding  terms ;  the 
scale  of  relation  consists  of  three  parts. 

389.  To  find  the  scale  of  relation  and  the  sum  of  a  recurring 
series  of  the  second  order. 

1st.  Let  a,  1),  c,  d,  represent  the  coefficients  of  any  four  consec- 
utive terms;  and  let  in,  nf  denote  the  scale  of  relation.  Then  from 
the  nature  of  the  scries,  we  have 

ma -\-nb  =  c  ) 
ml-{-nc  =.  d\ 

These  two  equations  will  determine  the  values  of  m  and  n. 
2d.  To  find  the  sum  of  the  scries,  denote  the  terms  of  the  series* 
by  A,  B,  C,  etc.,  and  let 

S'=  A+B+C+D+E+   ...  (1) 


RECURRING    SERIES.  333 

TLc  scries  is  supposed  to  contain  the  ascending  powers  of  x,  the 
first  power  occurring  in  either  A  or  B.  Then  because  the  series  is 
of  the  second  order,  we  have 

C  =  mAx*+nBx 

D  =  mBx*+nCx  ,0v 

E  =  mCx*+nDx 
etc.         etc.         etc. 
Adding  these  equations,  and  observing  the  value  of  S  in  (1),  we 
have 

S—A—B  =  mx*S+nx(S—  A). 
Whence  we  obtain 


39O.  To  find  the  scale  of  relation  and  the  sum  of  a  recurring 
series  of  tlie  third  order. 

1st.  Let  a,  b,  c,  rf,  e,J\  represent  the  coefficients  of  any  six  con- 
secutive terms ;  also  represent  the  scale  of  relation  by  m,  72,  r. 
Then  we  have, 

ma-\-nb-\~rc  =  d  \ 

ml)~\-nc-\-rd  =  e    >  (7*) 

mc-\-nd-\-re  =  f  ) 
These  three  equations  will  determine  the  values  of  m,  n  and  r. 

2d.  To  find  the  sum  of  the  series,  represent  the  terms  by,  A,  B, 
C,  etc.,  and  put 

S=A+B+C+D+E+F+   ...         (1) 
Then  because  the  series  is  of  the  third  order  we  shall  have 
D  =  mAx*+nBx*+rCx'\ 
E  =  m&tf+nCx*+rDxf 
F  =   mCx*+nDx*+rEx{ 
etc.,       etc.,       etc.,       etc.  y 
By  addition,  observing  the  value  of  S  in   (1),  we  have 

S—A—B—C  =  vnx'S+nx\S— A)+rx(S—  A— J5) 
we  obtain 

A+£+C—\ 


—--— 
1  —  rx  —  IMC*  —  nix* 

In   Hkv  manner  formulas  may  be  obtained  for  the  summation  of 
recurring  series  of  higher  orders. 


334  SERIES. 

391.  To  apply  these  formulas  in  the  summation  of  any  given 
series,  we  must  first  determine  the  scale  of  relation  by  (P)  or  (I7), 
and  then  we  may  obtain  the  sum  of  the  series  from  (Q)  or  (V). 

If  the  order  of  the  series  is  not  known,  we  should  first  deter- 
mine the  values  of  m  and  n  by  formula  (P).  and  ascertain  by  trial 
whether  the  scale  of  relation  thus  found  will  apply  to  the  givon 
series.  If  it  will  not  apply,  we  may  determine  the  values  of  m,  •//, 
and  V  from  formula  (T\  and  ascertain  by  trial  whether  the  series 
can  be  developed  by  the  new  scale  thus  obtained.  If  this  also  fail, 
we  must  establish  other  formulas  corresponding  to  still  higher  de- 
grees, and  continue  the  trials. 

If,  however,  we  resort  in  the  first  place  to  a  formula  corresponding 
to  an  order  higher  than  that  of  the  given  series,  then  one  or  more 
of  the  quantities,  m,  n,  r,'  etc.,  will  prove  to  be  zero,  and  the  re- 
maining numbers  may  be  taken  as  the  scale  of  relation,  without 
further  trial. 

1.  Find  the  sum  of  the  infinite  series,  1  -f-  4x  -f-  10xf  -j-  22x*  -f 
46x4-f  .... 
To  determine  the  scale  of  relation,  we  have 

a  =  1,  I  =  4,  c  =  10,  d  =  22. 
These  values  substituted  in  formula  (/*),  give 

m+4»  =  10, 
4w+10»  =  22, 
from  which  we  readily  obtain 

m  —  —2,     n  =  30 

These  numbers  form  the  true  scale  of  relation  ;  for  we  perceive 
that  any  coefficient  after  the  second  in  the  given  series,  is  equal  to 
three  times  the  first  preceding  coefficient,  minus  twice  the  second 
preceding  coefficient. 

To  find  the  sum  of  the  series,  we  have 

A  =  1,     B  =  4*. 
Whence  by  formula  (Q) 


+x 

~  -  ' 


We  have  "thus  obtained  the  sum  of  the  series  in  the  form  of  an 


RECURRING    SERIES 

algebraic  fraction.  Conversely,  the  given  series  may  be  developed 
from  this  fraction,  either  by  division,  or  by  the  method  of  indeter- 
minate coefficients.  Indeed,  it  will  be  found  that  the  sum  of  every 
recurring  scries  is  an  irreducible  fraction,  from  which  the  series 
may  be  supposed  to  originate.  The  fraction  from  which  any  partic- 
ular series  is  supposed  to  arise,  is  called  the  generating  fraction  for 
that  series  ;  it  is  the  same  as  the  sum  of  the  series. 


EXAMPLES    FOR   PRACTICE. 

1.  Find  the  sum  of  l+3*+4*«+7*'+ll*4+  .... 

1+2* 
Ans. 


2.  Find  the  sum  of  l+6*+12*'+48*'+120*4+ .... 

1+5* 
Ans.  -r- 


3.  Find  the  sum  of  1+2*— 5*'+2G*'— 119*4+ .... 

+  6.r 


Ans. 

4.  Find  the   sum  of  1  -f  4o;  +  3*'  —  2x*  +  4 x<  +  1 7ar*  +  3 *"  + 

^Jl* 

5.  Find  the  sum  of  l+3*+5*f+7*'+9*4+ 

Ans. 

(I— * 

6.  Find  the  sum  of  l+*+5**+13*>+41.r4+121*B+ 

Am. 


1—2*— 3** 

7.  Find  the  sum  of  l+4*+6**+ll*a+28*4+63*6+ 

Ans. 


x  7r*  fil  r* 

8.  Find  the  sum  of  -  +  x'+  '—  +  10*'+  -^-  +91*"+  . . . 

iu  a  u 


Ans.  - — r-=- 


336 


SEEIES. 


DIFFERENTIAL  METHOD. 


393.  Tlic  Differential  Method  is  the  process  of  finding  any 
term  of  a  regular  scries,  or  the  sum  of  any  number  of  terms,  by 
means  of  the  successive  differences  of  the  terms. 

303.    To  find  any  term  of  a  series  ly  the  differential  method. 

If  we  subtract  cacli  term  of  a  series  from  the  next  succeeding 
term,  we  shall  obtain  a  new  scries  called  t^Q  first  order  of  differences. 
If  we  subtract  each  term  of  this  new  scries  from  the  succeeding 
term,  we  shall  obtain  a  series  called  the  second  order  of  difference*  ; 
and  so  on. 

Let  r/,  &,  c,  d,  c,  ....  represent  a  regular  series,  the  successive 
terms  being  formed  according  to  any  fixed  law.  We  will  write  the 
given  terms  in  a  vertical  column,  and  proceed  by  actual  subtraction 
to  form  the  several  orders  of  differences,  placing  each  order  in  a 
separate  column,  and  each  difference  at  the  right  of  the  subtrahend. 
The  result  is  as  follows  : 


.Series. 

1st  order  of 
differences. 

2'1  onler  of 
differences. 

3d  order  of 
differences. 

4th  orler  of 
differences. 

a 

I 

&—  a 

c 

c—  I 

c_2i+a 

d 

d—c 

d—2  c-f-6 

rf_3c_|_3i_  a 

e 

c—d 

c—2d+c 

c—3d+Sc—l 

e_  4<?-f-Gc—  41-fa 

The  quantity  which  stands  first  in  any  column,  though  a  polyno- 
mial, is  called  the  first  term  of  the  order  of  differences  which 
designates  the  column. 

Let  c/f,  rf2,  <73,  rf4,  etc.,  represent  the  first  terms  of  the  first, 


second,  third,  fourth,  etc.,  orders  of   differences. 
have 

dl  =  b—a 


Then  we  shall 


rfs  =d— 

d4  =  e— 


—  46+a,  etc. 


DIFFERENTIAL    METHOD. 


By  transposition,    we    may   obtain,   after  making  the  necessary 
substitutions, 


c  =  a-f  4<7,-f  6<72-f  4r73-{-r74,  etc. 

These  equations  express  the  values  of  a,  b,  c,  d,  e,  etc.,  in  terms 
of  a,  c?,,  rf3,  r73,  <74,  etc.  The  coefficients  in  the  second  members 
are  formed  according  to  the  binomial  formula  ;  and  we  observe  that 
the  coefficients  of  the  second  power  of  a  binomial  are  found  in  the 
third  equation,  the  coefficients  of  the  third  power  in  the  fourth 
equatioo,  and  so  on. 

Hence,  if  we  let  Tn+l    denote  the   (ra-f-l)£/i  term  of  the  given 

series, 

a,  b,  c,  d,  e,  ----  . 

then  we  shall  have 


And  by  substituting  n  —  1  for  n,  we  shall  obtain  a  formula  for  the 
7ith  term  of  the  given  series  ;  thus, 

T*  = 

%_n(n—  2)  Oi—lXw 

l)cf  ,  +  i  ---   --  >-  d.2  +  V  - 


.    To  find  the  sum  of  any  number  of  terms  of  a  series,  by 
ike.  differential  method. 

Represent  the  given  series  by 

a,  b,  c,  d:  e,  .  .  .  .  (1) 

And  denote  the  sum  of  n  terms  by  S.     We  are  to  find  the  values 
of  S  in  functions  of 

a,  d19  dZJ  <73,  etc. 

,Lct  us  assume  the  auxiliary  series, 

0,     «.,     «-f-&,     tt-f6-|-c,     a-f6+c+rf,      ----         (2) 
it  is  ob^.ious  that  the  (n-(-l)th  term  of  this  series  is  the  same  as 
the  sum  of  n  terms  of  the  given  series,  (1),  and  may  be  placed  equal 
to  S.     Now  let 

c?V     </'„,     f7'3,     </'4,     etc., 
21)  w 


333  SERIES. 

represent  the  first  terms  of  the  successive  orders  of  differences  in 
the  auxiliary  series  (2).     Then  by  formula  (.4),  we  have 


But  if  we  proceed  to  form  the  first,  second,  third,  etc.,  orders  of 
differences  for  the  auxiliary  series  (2),  we  shall  have 
</',=«, 


rf'a  =  c—  2£-{-a  =  <?a,  etc. 
Hence,  by  substitution  in  equation  (n)  we  have 

nfr,  —  1)  v         n(n  —  1)  (»_2) 
S  =  na  +  A_J  rfi+    '.    ^     A_J  ^  _  _   (B) 

39«>.  The  use  of  formulas  (A)  and  (_£)  may  be  illustrated  by 
the  following  examples  : 

1.  Find  the  12th  term  of  the  series,  1,  5,  15,  35,  70,  126,  etc. 
We  first  form  the  successive  orders  of  differences,  as  follows  : 

1, 

5,      4, 
15,     10,       6, 
35,     20,     10,    4, 
70,    35,     15,     5,     1, 
126,     56,     21,     6,     1,    0. 
Thus  we  have  n  =  12,  and 

-    **  =  !,     d,=4,     </«  =  6,     ^3=4,     ^=1,     ds=Q. 
Substituting  these  values  in  (-4),  and  reducing  the  terms,  we  obtain 

T,,  =  l-f-44+330+660-f  330  =1365,  Am. 
The  series  is  broken  off  at  the  fifth  term,  because  the  subsequent 
differences  are  all  zero. 

2.  Sum  the  series  1,  3,  6,  10,  15,  21,  etc.,  to  n  terms. 

By  forming  the  successive  orders  of  differences,  as  in  the  iasi 
example,  we  shall  obtain 

a  =  l,  d,  =2,  rf3  =  l,  df3  =  0. 
Whence,  by  formula  (#), 


DIFFERENTIAL    METHOD.  339 


o  | 


all  the  terms  after  the  third  becoming  zero.      By  performing  the 
indicated  operations,  adding  the  results,  and  then  factoring,  we  have 


S  =  ,  A*. 


EXAMPLES     FOR    PRACTICE. 

1.  Find  the  9th  term  of  the  scries  1,  4,  8,  13,  19,  etc. 

Ans.  53. 

2.  Find  the  15th  term  of  the  scries  1,  4,  10,  20,  35,  etc. 

Ans.  680. 

3.  Find  the  8th  and  9th  terms  of  the  scries  1,  6,  21,  56,  126, 
251,  456,  etc.  Ans.  781  and  1231. 

4.  Find  the  20th  term  of  the  scries  1,  8,  27,  64,  125,  etc. 

Am.  8000. 

5.  Find  the  nth  term  of  the  series  1,  3,  6,  10,  15,  21,  etc. 

An,.  "^±1). 
z 

6.  Find  the  nth  term  of  the  scries  1,  4,  10,  20,  35,  etc. 


. 

7.  Find  the  nth  term  of  the  scries,  1,  5,  15,  35,  70,  126,  etc. 

n(n+l)(n+2)(n+3) 
~~24~ 

8.  Sum  the  series  1,  3,  6,  10,  15,  21,  etc.,  to  20  terms. 

Ans.  1540. 

9.  Sum  the  series  1,  5,  14,  30,  55,  91,  etc.,  to  12  terms. 

Ans.  2366. 

10.  Sum  the  series  1,  4,  13,  37,  85,  166,  etc.,  to  10  terms. 

Ans.  2755. 

11.  Sum  the  series,  1  •  2,  2  •  3,  3  •  4,  4  •  5,  5  •  6,  etc.,  to  n  terms 

n(n+l)fn+.2) 
Ans.   -        —-     —  , 


340  SERIES. 

12.  Sum  the  series  1-2-3,     2-3-4,    3-4-5,    4-5-6,  etc.,  to 

n  terms.  w(n-fl)(n-J-2)(w-|-3) 

Ans.  •  -  t 

4 

13.  Sum  the  scries  lf,  2a,  3*,  49,  5',  etc.,  to  n  terms. 

Am    "(»+l)(2»+l) 
6 

14.  Sum  the  scries  1',  2»,  38,  4*,  5',  etc.,  to  n  terms. 

.  £+=>V 

15.  Sum  the  series  I4,  24,  34,  44,  54,  etc.,  to  n  terms. 

n*       ?i4      n*        n 

^  ^+2~  +3  ~  sir 

16.  Sum  the  series  (m+1),  2(m+2),  3(m+3),  4(m-|-4),  etc., 
to  n  terms. 


INTERPOLATION. 

3OG.  Interpolation  is  the  process  of  introducing  between  the 
terms  of  a  series,  intermediate  terms  which  shall  conform  to  the 
law  of  the  series.  It  is  of  great  use  in  the  construction  of  mathe- 
matical tables,  and  in  the  calculations  of  Astronomy. 

397.  The  interpolation  of  terms  in  a  series  is  effected  by  the  dif- 
ferential method.  In  any  series,  the  value  of  a  term  which  has  n 
terms  before  it  is  expressed  by  formula  (wi),  (393),  which  is 

,    n(n—  1)   T          nn—  lYn—  -  2 
n  -       --  ;  d 


, 


If  in  this  formula  we  make  n  a  fraction,  then  the  resulting  equa- 
tion will  give  the  value  of  a  term  intermediate  between  two  of  the 
given  terms,  and  related  to  the  others  by  the  law  of  the  series. 

If  n  is  less  than  unity,  the  intermediate  term  will  lie  between  the 
first  and  second  of  the  given  terms  ;  if  n  is  greater  than  1  and  less 
than  2.  the  intermediate  term  will  lie  between  the  second  and  third 
of  the  given  terms  \  and  so  on. 


INTERPOLATION. 


341 


=  2.758924  , 

'22  =  2.802039  /  to  find  the  cube  roots 

Given       ^23  =  2.843807  /  of  intermediate  nuin- 

'24  =  2  88450Abers}  by, interpolation. 

.  '25  =  2.924018/ 
1.  Required  the  cube  root  of  21.75. 

We  have 


No. 

Cube  Boots. 

d, 

d, 

** 

d4 

21 

2-758924 

22 

2.802039 

+.043115 

23 

2.843867 

-[-.041828 

—.001287 

24 

2.884501 

+•040632 

—.001196 

+.000091 

25 

2.924018 

+.039519 

—.001113 

+.000083 

—.000008 

Hence,  to  find  the  cube  root  of  21.75  by  the  formula,  we  have 
a  =  2.758924,  n  =  .75, 

dl  =  +.043115,  d.2  -  —.001287,  d3  =  +.000091,  etc. 
These  values  substituted  in  the  formula,  give 
1st    term,      +2.758924 
"         +  .032336 
"         +  .000121 


2d 
3d 
4th 


Whence, 


+  .000004 
2.791885,  Arts. 


If  it  were  required  to  find  the  cube  root  qf  any  number  between 
22  and  23,  we  might  put  n  equal  to  the  excess  of  the  number 
above  21,  and  employ  the  same  values  for  c/,,  c?2,  t?3,  etc.,  as  before. 
But  greater  accuracy  will  be  attained  by  making  22  the  first  term 
of  the  series,  and  employing  the  corresponding  differences ;  in 
which  case  n  will  be  a  proper  fraction. 


EXAMPLES     FOR    PRACTICE. 


Find  by  interpolation, 

1.  The  cube  root  of  21.325. 

2.  The  cube  root  of  21.875. 

29* 


Ans.  2.773083. 
An*.  2.796722. 


342  SERIES. 

3.  The  cube  root  of  21.4568.  Ans.  2.778785. 

4.  The  cube  root  of  22.25.  Ans.  2.812613. 

5.  The  cube  root  of  22.684.  Ans.  2.830784. 

6.  The  cube  root  of  22.75.  Ans.  2.833525. 
398.  On  three  consecutive  days,  the  angular  distances  of   tho 

sun  from  the  moon,  as  seen  from  the  earth,  were  as  follows : 


1st  day,  noon, 

66°    6'  38". 

"      "     midnight, 

72°  24'    5". 

2d     "     noon, 

78°  34'  48". 

"      "     midnight, 

84°  39'    4". 

3d     "     noon, 

90°  37'  18". 

"     "     midnight, 

96°  29'  57". 

In  the  data  here  given,  the  interval  of  time  is  12  hours.  Hence, 
to  find  the  distance  of  the  sun  from  the  moon  at  intermediate  times, 
ft  must  always  be  some  fractional  part  of  12.  Thus,  for  the  distance 
at  3  o'clock  p.  M.  of  the  first  day  we  have  n  =  -fy  =  J,  and  a  =. 
66°  6'  38" ;  for  the  distance  at  6  o'clock  A.  M.  of  the  second  day, 
n  =  T«2  =  A,  and  a  =  72°  24'  5".  For  the  distance  at  3  o'clock 
p.  M.  of  the  second  day,  n  =  T\  =  |,  and  a  =  78°  34'  48". 


EXAMPLES  FOR   PRACTICE. 

Find  by  interpolation  the  distance  of  the  sun  from  the  moon, 

1.  At  3  o'clock  p.  M.  of  the  first  day.  Ans.  67°  41'  38" 

2.  At  6  o'clock  p.  M.  of  the  first  day.  Ans.  69°  16'  13" 

3.  At  9  o'clock  p.  M.  of  the  first  day.  Ans.  70°  50'  21" 

4.  At  3  o'clock  A.  M.  of  the  second  day.  Ans.  73°  57'  23" 

5.  At  6  o'clock  A.  M.  of  the  second  day.  Ans.  75°  30'  16" 

6.  At  9  o'clock  A.  M.  of  the  second  day.  Ans.  77°    2'  44" 

7.  At  3  o'clock  p.  M.  of  the  second  day.  Ans.  80°    6'  27", 

8.  At  6  o'clock  p.  M.  of  the  second  day.  Ans.  81°  37'  43", 

9.  At  9  o'clock  p.  M.  of  the  second  day.  Ans.  83°    8'  35" 


LOGARITHMS. 


LOGARITHMS. 


The  Logarithm  of  a  number  is  the  exponent  of  the  power 
to  which  a  certain  other  number,  called  the  Lage,  must  be  raised,  in 
order  to  produce  the  given  number.  Thus,  in  the  expression, 

<r=b, 

the  exponent,  x,  is  the  logarithm  of  l>  to  the  base  a. 

An  equation  in  this  form  is  called  an  exponential  equation. 

If  in  this  equation  we  suppose  a  to  be  constant,  while  b  is  made 
equal  to  every  possible  number  in  succession,  the  corresponding 
values  of  x  will  constitute  a  system  of  logarithms  :  hence, 

4OO.  A  System  of  Logarithms  consists  of  the  logarithms  of 
all  possible  numbers,  according  to  a  given  base. 

Any  positive  number  greater  than  unity  may  be  made  the  base 
of  a  system  of  logarithms.  For,  by  giving  to  x  suitable  values,  the 
equation  ax  =  b 

will  be  true  for  all  possible  values  of  &,   provided  a  is  positive  and 
greater  than  1.     Hence, 

There  ma./  be  an  indefinite  number  of  systems  of  logarithms. 

SOB.  If  in  the  equation  a*  =  b,  we  suppose  b  to  represent  a 
perfect  power  of  a,  then  x  will  be  some  integer  •  but  if  />  is  not  a 
perfect  power  of  «,  then  x  will  be  some  fraction.  Hence, 

A  logarithm  may  consist  of  an  integral  and  a  fractional  part. 

4O2.  The  Index  or  Characteristic  of  a  logarithm  is  the  integral 
part  5  and 

<1O«S.  The  Mantissa  is  the  fractional  part  of  a  logarithm. 
For  illustration,  let  5  be  the  base  of  a  system  ;  then  we  have 

52'25  =  5+  =  VV  =  37.384. 

Thus,  the  logarithm  of  37.384  to  the  base  5,  is  2.25  ;Vthe  index 
of  this  logarithm  is  2,  and  the  mantissa  .25. 

PROPERTIES  OF  LOGARITHMS. 

4LO4.  There  are  certain  properties  of  logarithms,  which  are 
common  to  all  systems.  To  investigate  these  general  properties,  let 


344  SERIES. 

a  denote  the  base  of  the  system ;  also,  designate  the  logarithm  of  a 
quantity  by  log.,  written  before  the  quantity. 

1. — In  am/  system,  the  logarithm  of  unity  is  0. 

For,  let  a*  =.  1  ]  then  x  ==.  log.  1. 

But  by  (88),  if  a*  =  1,  then  x  =  0,  or  log.  1  =  0. 

2. —  In  any  system,  the  logarithm  of  the  base  itself  is  unity. 

For,  let  a*  =  a  ;  then  x  r=  log.  a. 

But  by  (88),  if  a*  =  a,  then  x  =  1,  or  log.  a  =  1. 

3. —  The  logarithm  of  the  product  of  two  numbers  is  equal  to  the 
sum  of  the  logarithms  of  the  two  numbers. 

For,  let  MI  =  az,  n  =  a* ; 

then  x  =  log.  m,          z  =  log.  n. 

But  by  multiplication  we  have 

mn  =  a*+' ; 
therefore,  log.  mn  =  x-\-z  =  log.  m-j-log.  n. 

4. — The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the 
dividend  diminished  by  the  logarithm  of  the  divisor. 

For,  let  m  =  axy         n  =  a* ; 

then  x  =  log.  m,  z  =  log.  «.. 

By  division  we  have  _  —  a*-*  • 


n. 


therefore,  log.  (  —  j  =  x — z  =  log.  m — log. 

5. —  The  logarithm  of  any  power  of  a  number  is  equal  to  tht 
logarithm  of  the  number  multiplied  by  the  exponent  of  the  power. 

For,  let  m  =  a* ;  then  x  =  log.  m. 

By  involution  we  have          mr  =  an ; 
therefore,  log.  (mc)  =  rx  =  r  log.  m. 

6. —  The   logarithm   of  any    root  of   a  number  is  equal  to  the 
logarithm  of  the  numbei'  divided  by  the  index  of  the  root. 

For,  let  m  =  a* ;     then  x  —  log.  m. 

By  evolution  we  have 


LOGARITHMS. 

.  The  principal  use  of  logarithms  is  to  facilitate  arithmet- 
ical computations.  By  means  of  the  last  four  properties,  we  may 
avoid  the  ordinary  labor  of  multiplication,  division,  involution,  and 
evolution, — these  operations  being  practically  performed  by  addition 
and  subtraction. 

For  this  purpose,  it  is  necessary  to  have  a  ToUe  of  Logarithms, 
so  constructed  that  we  may  readily  obtain  the  logarithm  of  any 
numbqr  within  a  certain  limit,  or  the  number  corresponding  to  any 
logarithm,  to  a  certain  degree  of  approximation.  The  common 
tables  give  the  logarithms  of  numbers  from  1  to  10,000,  correct  to 
6  decimal  places. 

With  a  table  of  this  kind,  we  have  the  following  obvious 

RULES     FOR     COMPUTATION. 

I.  To  multiply  one  number  by  another  : — Find  the  logarithms 
of  the  given  numbers  ;  add  these  logarithms,  and  find  the  number 
corresponding  to  the  sum  ;  this  number  will  be  the  required  product ; 
(404,  3). 

II.  To  divide  one  number  by  another  : — Find  the  logarithms  of 
the  given  numbers  ;  subtract  the  logarithm  of  the  divisor  from  that 
of  the  dividend,  and  find  the  number  corresponding  to  the  difference  ; 
thi*  number  icill  be  the  required  quotient ;  (4O4,  4). 

III.  To  raise  a  number  to  any  power  : — Find  the  logarithm  of 
the  given   number,  and  multiply  it  by  the  exponent  of  the  required 
poirer  ;   then  find  the  number  corresponding  to  this  product,  and  it 
will  be  the  required  power  ;  (4O4.  5)- 

IV.  To  extract  any  root  of  a  number  : — Find  the  logarithm  of 
the  given  number,  and  divide  it  by  the  index  of  the  root ;  then  find 
the  number  corresponding  to  the  quotient,  and  it  trill  be  the  required 
ror>t;  (4O4,  6). 

NOTE. — From  (4OO),  we  infer  that  negative  numbers,  as  such,  have  no 
logarithms.  But  we  may  always  employ  logarithms  in  calculations  where 
negative  factors  are  involved,  by  disregarding  signs  until  the  absolute 
value  of  the  product  or  quotient  is  obtained. 


346  SERIES. 


THE    COMMON    SYSTEM. 

<1O6.  Any  positive  number  except  unity  may  be  made  tho 
base  of  a  system  of  logarithms.  But  the  only  base  used  in  practic- 
al calculations,  is  10.  The  logarithms  of  numbers  according  to 
this  base,  form  what  is  called  the  Common  System  of  logarithms. 

NOTE. — Besides  the  common  system,  there  is  another,  called  the  No- 
perian  System,  from  Baron  Napier,  the  inventor  of  logarithms.  This 
system  is  of  great  theoretical  importance,  and  its  relation  to  other  systems 
will  be  shown  in  a  subsequent  article. 

4O7.  The  peculiarities  which  constitute  the  advantage  of  tho 
common  system,  may  be  shown  as  follows  : 
Since  10  is  the  base  of  the  system, 

log.  1   =   log.  10°,  =  0, 

log.   10   =  log.  101   =   1, 

log.  100  =  log.  103  =  2, 

log.  1000  =  log.  103  =  3, 

log.  10000  =  log.  10*  =  4. 

Now  it  is  obvious  that  if  any  number,  integral  or  mixed,  be  great- 
er than  1  and  less  than  10,  its  logarithm  will  be  entirely  decimal; 
if  the  number  be  greater  than  10  and  less  than  100,  its  logarithm 
will  be  1  plus  a  decimal ;  if  greater  than  100  and  less  than  1000, 
its  logarithm  will  be  2  plus  a  decimal ;  and  so  on.  Hence, 

1. —  The  common  logarithm  of  an  integer  or  a  mixed  number 
icill  have  a  positive  index,  equal  to  the  number  of  integral  places 
minus  1 . 

Again,  since  the  logarithm  of  10  is  1,  it  follows  that  if  a  number 
be  divided  by  10  continually,  the  logarithm  will  be  diminished  by 
1  continually,  the  decimal  part  remaining  unchanged. 

Let  us  take  any  number,  as  5468,  and  denote  the  mantissa,  or  the 
decimal  part  of  its  logarithm,  by  m.  Then  we  have 

(1.)  (2.) 

log.  5468    =  3-fm,  log.  .5468         =  —  1-j-m, 

log.  546.8  =  2-f  w,  log.  .05468       =  —  2-fm, 

log.  54.68  =  1-fm,  log.  .005468     =  —  3-f-m, 

log.  5.468  ™  Q-fm;  log.  .0005468  =  —44m; 


LOGARITHMS.  ?>41 

in  which  3,  2,  1,  0,  are  the  indices  of  the  logarithms  in  column  (1); 
and  —  1,  —  2,  —8,  —  4,  are  the  indices  of  the  logarithms  in  column 
(2)  •  and  m,  the  decimal  part  in  all.  Hence, 

2.  —  If  two  numbers  consist  of  the  same  figures,  and  differ  only  in 
ike  position  of  the  decimal  point,  their  logarithms,  in  the  common  sys- 
tem, will  have  the  same  decimal  part,  and  will  differ  only  in  the  val- 
ues of  the  index. 

3.  —  The   common   logarithm  of  a  decimal  fraction  will  have  -a 
negative  index  ;  if  the  significant  part  of  the  decimal  commence  at 
the  tenths'  place,  the  index  of  the  logarithm  will  be  —  1  ;  but  if  ci- 
phers occur  between  the  decimal  point  and  the  first  significant  figure, 
the  index  of  the  logarithm  will  be  numerically  equal,  to   the  number 
of  intervening  ciphers,  plus  1. 

4O8.  In  writing  the  logarithm  of  a  decimal  fraction,  the  minus 
sign  is  placed  before  the  index,  and  the  decimal  or  positive  part  an- 
nexed without  any  intervening  sign.  Thus,  from  a  table  of  loga- 
rithms, we  have 

log.  .0546  =  —2.737193, 

in  which  the  minus  sign  must  be  understood  as  affecting  only  the 
index  2.  This  logarithm  is  therefore  equivalent  to 

—  2+.737193. 


COMPUTATION    OF   LOGARITHMS. 

4OO.  Since  the  rules  for  computing  by  logarithms  require  a 
logarithmic  table,  it  becomes  necessary  to  calculate  the  logarithms 
of  an  extended  series  of  "numbers.  The  only  practical  method  of 
doing  this,  is  by  means  of  a  converging  series,  expressing  the  value 
of  any  logarithm  in  known  terms. 

Let  us  resume  the  fundamental  equation, 

a*  =  b,  (1) 

in  which  x  is  the  logarithm  of  b,  to  the  base  a.  i 

Assume  a  ==  1  -j-c,  b  —  1-j-p  ; 

then  (1-M"  =  l+P,  (2) 

where  x  is  the  logarithm  of  1-f-p,  to  the  base  a. 


SERIES. 


Raise  both  members  of  equation  (2)  to  the  nth  power;  then 

(1+e)"*  =  (1-f  />)"• 
Expanding  both  members  by  the  Binomial  Theorem,  we  have 


«C«-l)(»-2)         «O-1)0—  2)Q—  3) 


Dropping  unity  irom  both  Tnembers,  and  dividing  by  ?i,  we  obtain 

/       (na-1)          (ng-l)(ng-2)  (wa-l)(na?-2X»g-8)  ,«  ,  \ 

aj^c"1        2  2-3  2.3.4  ") 

=  f  +  fi^+  (-^?/+  (-^^^-V+.  -  - 

This  equation  is  true  for  all  values  of  n  ;  it  will*  be  true,  there- 
,  fore  when  n  =  0.    Making  this  supposition,  the  eouation  reduces  to 

+*'  '- 


2  +  if  -4  +  5  -  •  •  •  -f  +     -      +--( 

From  equation  (2),  we  perceive  that 

'    *  ^  \  rr  --=  log.  (1+p). 

Hence,  if  we  place 

1 


equation  (3)  will  become 


Thus,  we  have  obtained  an  expression  Tor  the  logarithm  of  the 
number  l-}-p,  or  b.  This  expression  consists  of  two  factors;  namely, 
the  quantity  in  the  parenthesis,  which  depends  upon  the  number, 
.and  the  quantity  Jl/,  which  depends  upon  the  base  of  the  system. 

-41O.  It  is  obvious  that  if  a  definite  value  be  given  to  .If,  the 
base  of  the  system  will  be  fixed  and  determinate.  Baron  Napier 
arbitrarily  assumed  M=  1. 

To  determine  the  base  of  the  system,  according  to  this  assumption, 
substitute  1  for  M  in  equation  (4)  ;  (4rOO).  We  shall  have,  after 
reducing, 


LOGARITHMS.  349 


Putting  s  =  1,  we  shall  have 

c9         8        4 
*=C"  +  3 

.Reverting  the  series,  we  obtain 
«•  s' 


Restoring  the  value  of  s, 

=  l  +  F2  +  F^"3  +  1-2-3-4  +  1-2-3-4-5  +  *  *  '  ' 
By  taking  12  terms  of  this  series,  we  find  the  approximate  value 
of  c  to  be  1.7182818.     But  the  base  ^as  1-f-c;    hence,  adding  1  to 
the  resultyand  representing  thexsum.by  e,  the-  usual  symbol  for  the 
Naperian  base,  we  have 

e  =  2.T182818, 
which  is  the  base  of  the  Naperian  system.  • 

411.  In  the  general  formula,  (-4),  the  quantity  Jf,  which  depends 
upon  the  base,  is  called  the  modulus  of  the  system.  Thus,  the 
modulus  of  the  Naperian  system  is  unity. 

Let  us  here  designate  Naperian  logarithms  by  nap.  log.,  and  log- 
arithms in  any  other  system  by  log.,  simply.  Then, 


Dividing  (1)  by  (2),  we  obtain 

M=      I°g-(1+^)      .  (3) 

nap.  log.  (l+j>)  ' 

or,  {nap.  log.  (1+p)  JX^=  log.  (l+j>),  (4) 

where  M  is  the  modulus  of   the  system  in  which  the  logarithm  of 
the  second  member  is  taken.     Hence, 

The  modulus  of  any  particular  system  i»  the  constant  multiplier 
which  will  convert  Naperian  logarithms  into  the  logarithms  of  that 
system. 

30 


350 


SERIES. 


.  Formula  (A~)  can  be   employed   for   the  computation  of 
logarithms,  only  when  p  is  less  than  unity;  for  if  p  be  greater  than 
unity,  the  series  will  be  dwergintj.     The  series,  however,  may  be 
transformed  into  another  which  will  be  always  converyiny. 
Let  us  resume  the  logarithmic  series, 


If  in  this  equation  we  substitute  —  p  for  p,  we  shall 


If  we  subtract  equation  (2)  from  equation  (1),  observing  that 


we  sliall 


Assume 


These  values  substitu 


;  whence  we  obtain  „         = 

1— p — x  « 


^-  16  +  72*!'  +  '  '  /  ' 


4-  1)6       7(2*+!)       '  '    ' 
The  first  member  of  this  equation  is  equivalent  to  log.  (z-f-1)— 
log.  z.     Hence,  finally,  we  have  •  • 

—  log.3  = 


+l)7+"  /  : 


This  series  is  rapidly  converging,  and  may  be  employed  with 
facility  for  the  computation  of  logarithms,  in  the  Naperian,  or  in 
the  common  system. 

To  commence  the  construction  of  a  table,  first  make  z  =  1  ;  then 
log.  «  =  0,  and  the  formula  will  give  the  value  of  log.  (z-f-1),  or 
log.  2.  Next  make  z  =  2  ;  then  the  formula  will  give  the  value 
of  log.  (z-f  1),  or  log.  3  ;  and  so  on. 


LOGARITHMS. 


353 


It  is  necessary  to  compute  directly  the  logarithms  of  prime  num 
bers  only,  in  any  system;  for,  according  to  (404,  3),  the  loga 
rithm  of  any  composite  number  may  be  obtained,  by  adding  the 
logarithms  of  its  several  factors. 

413.  We  will  now  illustrate  the  use  of  formula  (J?)}  by  com- 
puting the  Nuperian,  logarithms  of  2,  4,  5,  and^lO. 

Make  2  =  1;  then  nap.  log.  2  =  0,  and  nap.  log.  (z-j-1)  =  nap 
log.  2  ;  and  since  M=  1,  we  have 


3  _       _  _.. 

We  first  form  a  column  of  numbers,  by  dividing  |  by  3s,  or  9, 
continually;  then  dividing  the  first  of  these  members  by  1,  the 
second  by  3,  the  third  by  5,  and  so  on,  we  obtain  the  several  terms 


of  the  series. 


32 


•    JL 


0 

.66666666  -r-  1  = 

.66666666 

7407407  ^  3 

2469136 

823045  -f-  5  == 

164609 

91449  -4-  7  = 

13064 

10161  -v-  9  = 

1129 

1129  -4-11  = 

103 

125  -j-13  = 

10 

14  -^15  = 

1 

.69314718  =  nap.  log.  2. 

2 

Whence,  by  (4O4,  5),  1.38629436  =  nap.  log.  4. 

Next  make  2  =  4;   then  2-j-l  =  5  ;   and  22-j-l  =  9  ;  and  we 


have 
nap. 


ioS.5=2(iL  +  1L.  +  ^L.  +  Ti-+....)+nap.lo,;.4. 


9 

=  81 
81 

81 

2 

0 

.22222222 
274348 
3387 
42 

+ 

1 

3 
5 

7 

= 

.22222222 
91449 
677 
6 

.22311354,  sum  of  series 


352  SERIES. 

To  .22314354 

Add  nap.  log.  4  =  1.38629436 

1.60943790  =  nap  log.  5. 
Add  nap.  log.  2=    .69314718 

Whence,  by  (4O4,  3),  2.8025?  508  =  nap.  log.  10. 

414.  In  order  to  compute  common  logarithms,  we  must  first 
determine  the  modulus  of  the  common  system.  From  (411), 
equation  (3),  we  have 


""  nap.  log.(l+p) 

In  this  equation,  make   1+p  =  10,  the  base  of  the  common 
system.     Then  we  have 


the  value  of  the  modulus  sought.  Substituting  this  value  in 
formula  (£*),  we  obtain  the  formula  for  common  logarithms,  as 
follows  : 

log.  (2+1) — log.  z  = 

(O 


To  apply  this  formula,  assume  z  =  10 ;  then 

log.  3  =  1,     and     23+1  =  21. 
.86858896 

^04136138  -f-  1  =  .04136138 
>    9379  +  3  =          3126 
®   21  +  5  =  4 


21 


441 


.04139268,  sum  of  scries. 

Add  log.  z  =  1.0 

log.  (2+1)  =  1.04139268  =  log.  11. 
If  we  make  z  =  99,  then  z+1  =  100,  and  2*+l  =  199. 
In  this  case,  the  formula  will  give  the  logarithm  of  99;  for, 
log.  (2+!)— log.  z  =  log.  100— log.  99  =  2—  log.  99. 


199 
199s  =  39601 


.80858896 

436477-7-1  =  .00436477 
11  -j-  3  =  4 


.00436481,  sum  of  series. 


LOGARITHMS. 


353 


Therefore,  we  have 

2— log.  99=    .00436481, 
whence,  1.99563519  =  log.  99. 

Subtract  log.  11  =  1.04139268 

.95424251  ''=  log.  9 
And  by  (4O4,  6),      4  log.  9  =    .47712126  =  log.  3. 

Thus  we  may  compute  logarithms  with  great  facility,  using  the 
formula  for  prime  numbers  only. 


USE   Or   TABLES. 

S 175.  The  following  contracted  tables  will  illustrate  the  princi- 
ples of  logarithms,  and  the  methods  of  using  the  larger  tables. 
The  logarithms  are  taken  in  the  common  system. 

TABLE  I. — LOGARITHMS  FROM  1  TO  100. 


H, 

Log. 

i  N< 

Log. 

N. 

Log. 

i  N. 

Log. 

1 

0  OWOOO 

26 

1  414973 

51 

1  707570 

i  76 

880814 

2 

0  301030 

27 

1  431364 

52 

1  716003 

77 

886491 

3 

0  477121 

28 

1  447158 

53 

1  724276 

78 

892095 

4 

0  602060 

29 

1  462398 

54 

1  732394 

79 

897627 

5 

0  698970 

30 

1  477121 

55 

1  740363 

80 

903090 

6 

0  778151 

i  31 

491362 

56 

1  748188 

81 

908485 

7 

0  845098 

32 

505150 

57 

1  755875 

82 

913814 

8 

0  908090 

33 

518514 

58 

1  763428 

83 

919078 

9 

0  954243 

i  34 

531479 

59 

1  770852 

84 

924279 

10 

1  000000 

35 

544068 

60 

1  778151 

85 

1  929419 

11 

1  041393 

36 

1  55f5303 

61 

1  785330 

86 

1  934498 

12 

1  079181 

37 

1  568202 

62 

1  792392 

87 

1  939519 

13 

1  113943 

38 

1  579784 

63 

1  799341 

88 

1  944483 

14 

1  146128 

39 

1  591065 

64 

1  806180 

89 

1  949390 

15 

1  176091 

40 

1  602080 

65 

1  812913 

90 

1  954243 

10 

1  204120 

41 

1  612784 

66 

1  819544 

91 

1  959041 

17 

1  230449 

42 

1  623249 

67 

1  826075 

92 

1  963788 

18 

1  255273 

43 

1  633468 

68 

1  832509 

93 

1  968483 

I  19 

1  278754 

44 

1  643453  ! 

69 

1  838849 

94 

1  973128 

20 

1  301030  T 

— 

45 

1  653213 

70 

1  845098 

95 

1  977724 

i  21 

1  322219  ! 

46 

1  602758  ' 

71 

1  851258 

08 

1  982271 

!  23 

T  342423 

47 

1  672098 

72 

1  857333 

97 

1  986772 

i  23 

1  -J  6  1  728 

48 

1  681241 

73 

1  863323 

98 

1  991226 

24 

1  380211 

49 

1  690196 

74 

1  869232 

99 

1  995635 

25 

1  397940 

50 

1  698970 

75 

1  875061 

100 

2  000000 

30* 


x 


854 


SEBIES. 


TABLE  II. — LoGAiUTinis  OF  LEADING  NUMBERS  WITHOUT  INDICES. 


JS. 

0. 

1. 

o 

3.  1  4. 

,5. 

6. 

7. 

8. 

9. 

100 

000000 

000434 

000868 

001301:001734 

002160 

002598 

003029 

003461 

008891 

101 

0043'21 

004750 

005181 

005609 

006038 

006466 

006894 

007321 

007748 

008174 

102 

008600 

C09026 

00945$ 

00987ti 

0103CO 

010724 

011147 

011570 

011993 

012415 

108 

012837 

013251; 

013680 

014100 

014521 

01494C 

015300 

015779 

016197 

016616 

104 

01703J-- 

017451 

01780K 

018284 

0187CO 

01911(1 

019532 

019947 

020o6.1 

020775 

105 

021189 

021603 

0220  Hi 

022428 

022841 

023252 

02301)4 

024075 

0244  8C 

024896 

100 

025306 

02571,1 

02G125 

020538 

026942 

027351 

027757 

028164 

028571 

028978 

107 

02938, 

02978H 

030195 

080600 

031004 

03140fc 

031812 

Oo2216 

032;;  if 

OS3021 

108 

033424 

033826 

03-122^ 

034628 

035029 

03543C 

035830 

036230 

036621 

037028 

109 

037420 

037825 

038223 

038620 

039017 

039414 

039811 

040207 

04060!: 

040998 

In  table  I,  the  logarithms  are  given,  with  indices,  in  columns 
adjacent  to  the  columns  of  numbers. 

In  table  II,  each  figure  in  the  row  at  the  top  may  be  annexed  to 
any  number  in  the  left-hand  column  ;  the  logarithm  of  any  number 
thus  formed,  will  be  found  at  the  right  of  the  number  in  the 
column,  and  beneath  the  figure  at  the  top.  The  proper  index  may 
be  supplie  1  in  any  case,  according  to  the  theory  of  logarithms. 
Thus,  to  obtain  the  logarithm  of  1023  by  this  table,  we  find  102  in 
the  left-hand  column,  and  3  in  the  top  row  j  and  opposite  the 
former,  and  under  the  latter,  we  'find  009876,  the  decimal  part  of 
the  logarithm.  Hence,  log.  1023  =  3.009876. 

In  like  manner,  we  find 

log.  104.2  =  2.017868,    log.  .1078  =  —1.032619. 

CASE   I. 

416.  To  find  the  logarithms  of  numbers  when  their 
factors  are  in  the  tables. 

RULE. —  Take  out  from  the  tables  the  logarithms  of  the  factors, 
and  find  their  sum  ;  the  result  will  be  the  logarithm  required. 


EXAMPLES   FOR   PRACTICE. 

1.  Required  the  logarithm  of  533.5. 
Observe  that  533.5  =  106.7x5 ; 

hence,  log.  106.7  =  2.028164 

log.  5         =    .698970 

2.727134,  AM. 


LOGARITHMS.  355 


2.  Find  tho  logarithm  of  520.  'tfjAm.  2.710003 

3.  Find  the  logarithm  of  146.  f  3  X  ?-  Ans.  2.164353. 

4.  Find  the  logarithm  of  1450.  if+^  +  f*      Am.  3.161368. 

5.  Find  the  logarithm  of  1.59.  '^l^^^l^Ans.  .201397. 

6.  Find  the  logarithm  of  2034.  (ifj  y  ^       Ans.  3.308351. 

7.  Find  the  logarithm  of  76.37.  SO-  ?/  *7        Ans.  1.882923. 

8.  Find  the  logarithm  of  .0201..^**^  *-    .4ns.  —2.303196.^*7* 

9.  Find  the  logarithm  of  .3822.  .f/  A-4  2-     Ans.  —  1.582290.  /."/£/'  v 
10.  Find  the  logarithm  of  16995./0-3/  33+f-    Ans.  4.230321. 


417.  To  find  the  logarithms  of  numbers  intermediate 
between  the  numbers  on  the  table. 

Since  the  logarithms  in  any  table  form  a  regular  series,  we  may 
interpolate  for  intermediate  logarithms,  by  the  usual  formula, 


If  the  logarithm  of  the  given  number  is  intermediate  between 
the  logarithms  of  table  I,  it  will  be  necessary  to  take  account  of  the 
first  and  second  differences.  But  we  may  always  employ  table  II, 
where  the  logarithms  increase  so  slowly  that  two  terms  of  the 
formula  will  give  the  result  accurately. 

The  first  four  figures  of  a  number,  counting  from  the  left,  will 
be  called  the  four  superior  figures  ;  and  the  others,  the  inferior 
figures.  To  apply  the  formula,  a  will  represent  that  logarithm  of 
the  table  which  is  next  less  than  the  required  logarithm,  and  n' 
will  denote  the  inferior-  figures  of  the  number,  regarded  as  a 
decimal. 

Hence  the  following 

RULE.  —  Take  out  the  logarithm  of  the  four  superior  figures  of 
the  given  number  ;  multiply  the  difference  between  this  logarithm 
and  the  next  greater  in  the  table,  by  the  inferior  places  of  the  num- 
ber, considered  as  a  decimal;  add  this  product  to  the  former  resulf, 
<md  the  sum  will  be  the  logarithm  required. 


356 


SERIES. 


EXAMPLES    FOR    PRACTICE. 

1.  Required  the  logarithm,  of  1.07632. 

This  number  is  found,  between  1.076  and  1.077;  hence 

log.  1.077— log!  1.076  =  404  =  dt. 
And  putting  n  =  .32,  we  have 

a  =  log."  1.076  =  .031812 
ndi  =  404  X  »32  =          129 

.031941,  Ans. 

2.  Required  the  logarithrn*of  3579. 

In  order  to  make  use  of  table  II,  we  proceed  thus : 

3579  -j-  35  =  102.25714-j-. 
log.  102.2— log.102.3  =  425  ;    n  =  .5714 
log.  102.2  =  2.009451 
425  X -5714=          243 

2.009694  =  log.  102.25714 
log.  35  -  1.544068 

,3.553762,  Ans. 

NOTE. — It  is  obvious  that  if  we  divide  any  number  by  its  first  two 
figures,  we  may  obtain  the  logarithm  of  the  quotient  by  means  of  table 
II ;  then  we  may  add  the  logarithm  of  the  divisor,  found  by  Table  I,  tc 
obtain  the  required  logarithm. 

3.  Find  the  logarithm  of  10724.  Ans.  4.030857. 

4.  Find  the  logarithm  of  10.8539.  Ans.  1.035586. 

5.  Find  the  logarithm  of  1021.56.  Ans.  3.009264. 
F6     6.  Find  the  logarithm  of  568.53.  4*^*Jf»«f4**  2.754753. 

+3        7.  Find  the  logarithm  of  3244.  [#//;///  Ans.  3.511081. 

J-M6    8.  Find  the  logarithm' of  365.25638.  Am.  2.562598. 

9.  Find  the  logarithm  of  132.57.#I//.£;3  Ans.  2.122445. 

_10.  Find  the  logarithm  of  567521.  Ans.  5.753982. 

Find  the  logarithm  of  258.7. ^V^j^  Ans.  2.412796. 

+  IZ  12.  Find  the  logarithm  of  1.296.  Ans.  .112605. 

J-/-7  13.  Find  the  logarithm  of  5784.  Ans.  3.762228. 


»  a*  *. 


LOGARITHMS,  357 


EXPONENTIAL   EQUATIONS. 

418.  We  will  now  illustrate  the  application  of  logarithms  to  the 
solution  of  exponential  equations. 

1.  Given  2Z  =  10,  to  find  the  value  of  x. 

Suppose  the  logarithms  of  both   members  of  the  equation  to  be 
taken.     We  shall  have,  by  (4O4,  5), 

a;  log.  2  =  log.  10; 


2 

2.  Given  b*  =  $,  to  find  the  value  of  x. 


Raising  both  members  of  the  given  equation  to  the  power  denoted 
by  x,  we  have 


Taking  the  logarithms  of  both  members, 

log.  25  =  x  log.  3  —  x  log.  7  ;  whence, 

log.  25  1.897940 


x  = 


JL  =  —  3.79899+,  Ans. 


log.3— log.7  ~  .477121— .845098 
3.  Given  ra*  —  fifc,  to  find  the  value  ot  x. 
Taking  the  logarithms  of  both  members  of  the  equation  ,we  have, 
by  (4O4,  3  and  5). 

log.  r+x  log.  a  =  2  log.  b  +  log.  c ; 

2  log.  b  +  log.  c— log.  r 

whence,  x-=. ; —  • 

log.  a 

EXAMPLES    FOR   PRACTICE. 

1.  Given  7*  =  8,  to  find  the  value  of  x.      Ans.  x  =  1.06862. 

^ 

2.  Given  5*  =30,  to  find  the  value  of  x.     Ans.  x  —  .94640. 

3.  Given  a*  =  6V,  to  find  the  value  of  x. 

2  log.  £+3  log.  c 

Ans.  x  =  — — 

log.  a 


358  SERIES. 

4.  Given  = —  =  m.  to  find  the  value  of  x. 

a 


«  , 

log.  6 


5.  Given  ma*  =  &,  to  find  the  value  of  a. 

log.  a 


Ans.  x  — 


log.  1) — log.  m 

C.  Given  a'-}- lr'  —  ^c  and  a*  —  bj=2d,  to  find  rr  and  y. 

]oo\  (c  +  rf)          loo-,  (c—fl?) 

Ans.  x=-  — +,  y  =  .   "    v      — ;. 

log.  a  log.  A 

!_ 

7.  Given  729*  =  3,  to  find  the  value  of  x.  Ans.  x  =  6. 

8.  Given  216T=  12,  to  find  the  value  of  x.  Ans.  x=  .— ^— . 

3  g<   1J 

9.  Given  516*  =  12,  to  find  the  value  of  x. 

3  log.  43 

-nr+3- 
°4*n7^ 

10.  Given  6*  =  ~    ' \    J  *  to  find  the  value  of  x. 
fl 

18  log.  24-|-log.  17—3  log.  71 
-  -3  log.  G 


PROPERTIES    OF   EQUATIONS.  S59 


SECTION   VIII. 

PROPERTIES    OF  EQUATIONS. 
419.  Let  us  assume  the  equation, 


-2-f  ____  -|-  Tx-\-  U  =  0,  (1) 
in  which  m,  the  exponent  of  the  degree,  is  a  positive  whole  number. 
An  equation  not  given  in  this  form  may  be  readily  reduced  to  it,  by 
transposing  all  the  terms  to  the  first  member,  arranging  them  accord- 
ing to  the  descending  powers  of  the  unknown  quantity,  and  dividing 
through  by  the  coefficient  of  the  first  term. 

In  this  equation  the  coefficients,  ^4,  /?,  (7,  etc..  may  denote  any 
quantities  whatever;  that  is,  they  may  be  positive  or  negative,  entire 
or  fractional,  rational  or  irrational,  real  or  imaginary. 

The  term  U  may  be  regarded  as  the  coefficient  of  ,r°,  and  is  called 
the  absolute  term  of  the  equation. 

420.  If  the  equation  contains  all  the  entire  powers  of  jc,  from 
the  ?nth  down  to  the  zero  power,  it  is  said  to  be  complete;  if  some 
of  the  intermediate  powers  of  x  are  wanting,  it  is  said  to  be  incom- 
plete.    An  incomplete  equation  may  be  made  to  take  the  form  of  a 
complete  equation,  by  writing  the  absent  powers  of  x  with  ±0  for 
their  coefficient. 

421.  It  has  been  shown    (3Oo  )   that  any  expression  of  tho 
second  degree  whatever,  containing  but  one  unknown  quantity,  may 
be  resolved  into  two  binomial  factors  of  the  first  degree  with  respect 
to  the  unknown  quantity,  —  the  first  term  in  each  factor  being  this 
quantity,  and   the  second   term   one  of    the    roots  (with  its  sign 
changed)  of  the  equation  which  results  from  placing  the  expression 
equal  to  zero.     We  therefore  conclude  that  every  expression  of  the 
second  degree  may  be   regarded   as  the  product  of   two  binomial 
factors  of  the  first  degree. 

So  likewise  the  product  of  three  binomial  factors  of  the  first 
decree  with  respect  to  any  unknown  quantity,  will  be  an  expression 


360  PROPERTIES    OF    EQUATIONS. 

of  the  third  degree,  and  we  readily  see  that  by  varying  the  values 
of  the  second  terms  of  the  factors,  corresponding  changes  are  pro- 
duced in  the  product.  Thus, 

(x— 2)(z-}-3)(x— 5)  =  x8— 4x8— llx+  30, 
and,      (x— 2-f  l/^3)(x— 2— V~Z}(x+  J)  =  x8—  |xa-f  «f  ; 
also,       (x-f  1—  T/^SXx+l+V^Xx— 2)  =  x8— 8. 

From  these  and  other  examples,  which  may  be  increased  at  pleasure, 
it  is  inferred  that  any  expression  whatever  of  the  third  degree 
would  result  from  the  multiplication  of  some  three  factors  of  the 
first  degree  in  respect  to  x.  And  in  general,  any  expression  of  the 
with  degree  with  respect  to  its  unknown  quantity,  may  be  regarded- 
as  the  result  of  the  multiplication  of  m  binomial  factors  of  the  first 
degree  with  respect  to  that  unknown  quantity. 

422.  If  then  we  have  any  equation  formed  by  placing  a  poly- 
nomial containing  the  unknown  quantity,  x,  equal  to  zero,  and  we 
discover  the  binomial  factor  x — a  in  the  first  member,  it  is  evident 
that  a  is  a  root  of  the  equation ;    for,  when  substituted  for  x,   it 
reduces  the  first  member  to  zero. 

If  we  can  succeed,  therefore,  in  discovering  the  binomial  factors 
of  the  first  degree,  of  the  first  member  of  any  equation,  the  roots 
of  the  equation  will  be  the  values  of  x  obtained  by  placing  each  of 
these  factors,  successively,  equal  to  zero. 

This  reverse  process  of  resolving  the  first  member  of  an  equation 
into  its  binomial  factors  of  the  first  degree,  is  one  the  difficulty  of 
which  increases  rapidly  with  the  degree  of  the  equation;  and 
algebraists  have  as  yet  discovered  no  general  method  for  effecting 
this  resolution  for  those  of  a  higher  degree  than  the  fourth.  By 
special  processes,  however,  the  roots  of  numerical  equations  may  be 
found  exactly,  when  commensurable,  and  to  any  degree  of  approxi- 
mation when  not  commensurable. 

423.  In  order  to  discover  the  law  which  governs  the  product 
of  any  number  of  binomial  factors,  such  as  x-j-a,  x-f-&,  x-j-r,  etc., 
having  the   first  term  the  same  in  all,  and  the  second  terms  differ- 
ent, let  us  first  obtain  the  product  of  several  of  these  factors  by 
actual  multiplication  ;  thus. 


PROPERTIES   OF   EQUATIONS, 


x  -\-a 


X   -\-C 


+ar 

-fie 


ub  .r-f  abc  } 


=  (x+a)  Cr-j-&)  (.r-f  c) 


x''-\-<i 

-H 


-far/ 
+ld 


-f  «a/ 


L 


From  an  examination  of  these  several  products  we  arrive  at  the 
following  conclusions  : 

1.— The  exponent  of  the  leading  letter,  .r,  in  the  first  term  is 
equal  to  the  number  of  binomial  factors  used,  and  this  exponent 
decreases  by  l,from  term  to  term,  towards  the  right,  until  we  come 
to  the  last  term,  in  which  it  is  0. 

2. — The  coefficient  of  the  first  term  is  1 ;  that  of  the  second, 
the  sum  of  the  second  terms  of  the  binomial  factors ;  that  of  the 
third,  the  sum  of  all  the  different  products  formed  by  multiplying, 
two  and  two,  the  second  terms  of  the  binomial  factors  j  that  of  the 
fourth,  the  sum  of  all  the  different  products  formed  by  multiplying, 
three  and  three,  the  second  terms  of  the  binomial  factors ;  the  last, 
or  absolute  term,  is  the  continued  product  of  the  second  terms  of 
the  binomial  factors. 

It  might  be  inferred  from  what  has  been  now  shown,  that  however 
great  the  number  of  binomial  factors  employed,  the  coefficient  of 
that  term  of  the  arranged  product  which  has  n  terms  before  it,  would 
be  the  sum  of  all  the  different  products  that  cau  be  formed  by  mul- 
tiplying the  second  terms  of  the  binomial  factors  in  sets  of  n  and  n. 

Assuming  that  the  above  law  holds  true  for  a  number  ??i,  of  bi- 
nomial factors,  if  it  can  be  proved  that  it  still  governs  the  product 


JitiU  PKOPERTIES    OF    EQUATIONS 

when  an  additional  factor  is  introduced,  it  will  be  established  in  all 
its  generality.     Let  us  suppose  then,  that  in  the  product, 


U 


of  the  m  binomial  factors  x-\-u,  x-\-l,.  .  .  .,  .r-f-p,  the  law  of  form- 
ation is  the  same  as  that  found  by  the  actual  multiplication  of 
pcveral  factors. 

Introducing  the  factor  x-j-g-,  we  have 


x+q 


-fcf 


+43 


-f  Ar 


,.m-n+  I 


+  ....  +  U 


It  is  at  once  seen  that,  in  this  new  product,  the  law  in  respect  to 
the  exponents  is-  unbroken.  As  to  the  coefficients,  that  of  the  first 
term  is  still  1 ;  that  of  the  second  term  is  A-\-q;  and  since  .4  is  the 
sum  of  the  second  terms  of  the  m  factors  in  the  assumed  product, 
A-\-q  is  the  sum  of  the  second  terms  of  the  w-f-1  binomials.  The 
coefficient  of  the  third  term  is  B-\-Aq.  Now  B  is,  by  hypothesis, 
the  sum  of  the  different  products  of  the  second  terms,  of  the  m 
binomial  factors,  taken  two  and  two  in  a  set,  and  Aq  is  all  of  the 
additional  products  to  which  the  introduction  of  the  factor  .c  +  q 
can  give  rise  ;  hence  B-\-Aq  is  the  sum  of  all  the  products, taken  two 
and  two,  of  the  second  terms  of  the  m-|-l  binomials.  And  the 
coefficient  of  the  general  term,  that  is  the  coefficient  of  the  term 
having  n  terms  before  it,  is  N-{-Mq ;  but -N  is  the  sum  of -all  the 
products  of  the  second  terms,  taken  n  and  n,  of  the  binomial  factors 
which  enter  the  assumed  product  ]  and  because  M  is  the  sum  of  all 
the  products  of  these  second  terms,  taken  n — 1  and  n — 1,  Mq  is  the 
sum  of  all  the  additional  products,  taken  n  and  n,  which  can  result 
from  the  introduction  of  the  factor  .r-j-y. 

Now  we  have  proved,  by  actual  multiplication,  that  the  law  of  the 
product,  admitted  to  be  true  for  n  binomial  factors,  is  true  for  four 
factors  ;  hence  by  what  has  just  been  demonstrated,  it  is  true  for 
five  factors  ;  and  being  true  for  five,  it  must  be  true  for  six,  and  so 
on.  Therefore  the  law  is  general. 


PROPERTIES    OF    EQUATIONS.  363 

The    composition  of   the  coefficients   of  an   equation   in 
terms  of  its  roots. 

Let  us  take  any  number,  m,  of  binomial  factors,  as  x  —  a,  x  —  b, 
x  —  c,  .  .  .  .x  —  p,  x  —  <?,  in  which  a,  b,  c,  etc.,  mny  represent  any  quan- 
tities whatever.  Now  it  has  been  shown  (428)  that  the  contin- 
ued product  of  these  factors,  arranged  according  to  the  desending 
powers  of  x,  will  be  of  the  form, 

xn+Ax™-1  -j-.&c'—  «  -f  Oc~-*-f  ____  £c9+ 
in  which 

A  =  —  a  —  b  —  c  —  .  .  .  .  —  p  —  q, 

B  =  -\-ab-\-ac-\-ltc-\-  ....  -\-op-\-aq, 

C  =  —  a&c  —  6cc?  —  ace?  —  .  .  .  .  —  alp  —  aij, 


S  =  ±alcd.  .  .pqm_<t±l>cde.  .  ..p^m-odt  etc., 
^1=  ^fabcde.  .  .pqrH^l  ^bcdef.  .  .pqn^l  +  etc., 
U  '  =  ±abed.  .  .  -pqmj 

the  subscript  expressions  m  —  2,  m  —  1,  m,  denoting  the  number  of 

literal  factors  which  enter  each  term. 
We  thus  have  the  identical  equation, 


) 
j  ~ 


-f  U 

and  placing  the  second  member  of  this  equal  to  zero  we  have 
xm+^xm-1  -^Ex"*--  -f-  ____  Sx*+  Tx+  U  =  0         (2) 

an  equation  of  which  a,  6,  c,  .  .  .  .  jp,  ^  are  the  roots,  since  these 
values  substituted  in  succession  for  x  in  the  first  member  of  the 
eq.  (1)  will  cause  this  first  member,  and  consequently  the  second 
member,  to  vanish.  The  relations  between  the  coefficients  At  B,  C, 
etc.,  and  the  roots  of  eq.  (2),  may  be  expressed  as  follows: 

1  —  The  coefficient  of  the  second  term   is  equal  to  the  algebraic 
xum  of  all  the  roots,  with  the  signs  changed. 

2.  —  The  coefficient  of  the  third  term  is  equal  to  the  algebraic  sum 
of  all  the  different  products  formed  by  multiplying  the  roots,  two  and 
two. 

3.  —  The  coefficient  of  the  fourth  term  is  equal  to  the  algebraic 
mm  of  all  the  different  products  formed  by  multiplying  the  roots 
with  their  signs  changed,  three  and  three. 


364  PROPERTIES    OF    EQUATIONS. 

4,  —  And  in  general  ;    The  coefficient  of  the  term  having  n  terms 
before  it,  is  equal  to  the  algebraic  sum  of  all  the  different  products 
formed  by  multiplying  the  roots,  with  their  signs  changed  if  n   ?'.-: 
odd,  n  and  n.       Hence, 

5.  —  The  absolute  term  is  the  continued  product  of  all  the  roots, 
with  their  signs  changed  when  the  number  denoting  the  degree  of 
the  equation  is  odd. 

This  principle  will  enable  us  to  construct  an  equation,  the  roots 
of  which  are  given,  and  the  composition  of  eq.  (1)  shows  that  eq. 
(2)  thus  constructed  can  have  no  other  than  the  assumed  roots  ;  for 
there  is  no  value  of  x  differing  from  one  of  these  roots  which  can 
cause  the  first  member  of  eq.  (1)  to  disappear. 

From  this  we  might  conclude  that  every  equation  involving  but 
one  unknown  quantity,  has  as  many  roots  as  there  are  units  in  the 
exponent  of  its  degree,  and  can  have  no  more. 

4^*5.  Admitting  that  every  equation  containing  but  one 
unknown  quantity  has  at  least  one  root,  real  or  imaginary,  it  may 
be  demonstrated  that  the  first  member  of  every  equation  of  the 
with  degree,  the  second  member  being  zero,  may  be  regarded  as  the 
continued  product  of  m  binomial  factors  of  the  first  degree  with 
respect  to  the  unknown  quantity.  We  will  first  prove  that, 

If  &  is  a  root  of  an  equation  of  the  form 

x^Ax^1-^-  B.x~~*+  ....  Tx+U=  0,  (1) 

its  first  member  can  be  exactly  divided  by  x  —  a. 

For  if  we  apply  the  rule  for  division,  we  shall  finally  arrive  at  a 
remainder  which  will  not  contain  x  j  since  for  each  quotient  term 
obtained,  the  new  dividend  is  at  least  one  degree  lower  than  that 
which  precedes. 

Calling  the  entire  quotient  Q  and  the  remainder  R,  we  shall  have 


an  identical  equation.  The  substitution  of  a  for  x  causes  the  first 
member,  and  also  the  first  term  in  the  second  member  of  this  equa- 
tion, to  vanish.  Hence,  R  =  0.  But  by  hypothesis  R  does  not 
contain  x  ;  it  is  therefore  equal  to  zero  whatever  value  be  attributed 
to  x,  and  the  division  is  exact 


PROPERTIES    OF    EQUATIONS.  365 

4  2G.  The  converse  of  the  last  principle  is  also  true;  that  is, 

Jf  the  first  member  of  the  equation, 

xm+Axm-l-\-£xm-*-}-  ____  Tx+  £7=0, 
can  Ic  exactly  divided  ~by  x  —  a,  then  a  is  a  root  of  the  equation. 

For,  suppose  the  division  performed,  and  that  the  quotient  is  Q; 
then  we  shall  have  the  identical  equation, 

xm-\-Axm-1+JBxm-*  +  ____  Tx-\-  U  =  Q(x  —  a). 

But  x  =  a  causes  the  second  member  of  this  equation  to  vanish  ; 
it  will  therefore  cause  the  first  member  to  vanish,  and  consequently 
satisfy  the  given  equation. 

42  7»  Every  equation  containing  but  one  unknown,  quantity  has 
a  number  of  roots  denoted  l>y  the  exponent  of  its  degree,  and  no 
more. 

Resuming  the  equation, 


and  admitting  that  it  has  one  root,  o,  x  —  a  must  be  a  factor  of  its 
first  member  ;  (425).  The  quotient  which  arises  from  the  division 
of  the  polynomial, 

xm+Axm~1-}-  ____  Tx-\-  U, 

by  x  —  a,  will  be  of  the  form 

' 


we  shall  therefore  have  the  identical  equation, 


Now  the  second  member  of  this  equation  will  vanish  for  any 
value  of  x  which  reduces  the  second  factor  to  zero. 
If  then  the  assumed  root  of  the  equation, 

x»-l+A'x~-*+  ....  Fx+  U'  =  0, 
be  denoted  by  &,  we  shall  have 

«r~l+A'x~-*+  I        ,       7  ,  (xm~^A"x^  '+  ....  T"x+  U"\ 
...T'x+U'    \ 

A  third  equation  may  be  formed  in  the  same  way,  and  then  a 
fourth,  and  so  on,  until  the  (in  —  l)th  equation  is  finally  reached,  in 
which  the  second  factor  in  the  second  member  is  of  the  first  degree 
with  respect  to  x. 
35* 


366  PBOPEliTIES    OF    EQUATIONS. 

Taking  this  last  equation,  and  substituting  for  its  first  member  the 
second,  in  the  next  preceding  equation,  and  thus  continuing  the 
process  of  substitution  until  the  first  equation  of  the  scries  is  arrived 
at,  the  result  will  be  the  following  identical  equation  : 


)  (.r-«)(x-^)(x— c).... 

J    =  = 


....Tx+V  -   (         (*-?)(*- 2) 

The  second  member  of  this  equation  vanishes  for  any  one  of  the 
in  values, 

x  =  a,  x  =  6,  x  =  c, . . . .  x  =  p,  x  =  q, 

and  consequently  these  values  are  severally  roots  of  the  equation, 

Moreover,  no  value  of  x  that  differs  from  some  one  of  these 
values,  can  satisfy  the  equation ;  for  no  such  value  will  cause  any 
one  of  the  factors  in  the  second  member  of  the  identical  equation 
to  be  zero,  a  condition  requisite  to  make  the  product  zero.  The 
equation  therefore  has  ra  roots  and  no  more. 

428.  From  the  foregoing  principles  we  conclude, 

1. — That  in  an  equation  in  which  the  second  term  does  not  ap- 
pear,— that  is,  the  term  containing  the  next  to  the  highest  power  of 
the  unknown  quantity, — the  algebraic  sum  of  the  roots  is  0. 

2. — If  an  equation  has  no  absolute  term,  at  least  one  of  its  roots  is  0. 

3. — The  absolute  term  being  the  continued  product  of  all  the 
roots  of  an  equation,  it  must  be  exactly  divisible  by  each  of  them. 

4. — An  equation  may  be  constructed,  which  shall  have  any 
assumed  roots. 

5. — The  degree  of  an  equation  may  be  reduced  by  1  for  each  of 
its  known  roots. 

EXAMPLES. 

1.  What  is  the  equation  having  -f-2,  — 3  for  its  roots  ? 

Ans.  x*-\-x — 6  =  0. 

2.  What  is  the  equation  having  the  roots  -(-1?  — 2,  —4  ? 

Ans.  oj'-f-Sx'H-  2x— 8  =  0. 

3.  What  is  the  equation  having  for  its  roots  -j-3,  — 2,  — 1,  -|-5  ? 

Ans.  x4— 5x»— 7xa+29x+30  =  0. 


PROPERTIES  OF  EQUATIONS.  367 

4.  What   is   the   equation   of   which   the    roots  are    1-fV  —  5, 
1—1/^5,  -f  v'5,  —  !/5  ?       .4/zs.  x-4—  2x3-fxa4-10.r—  30  =  0. 

5.  What  is  the  equation  of  which  the  roots  are  —  1,  —  2,  -|-3, 
2+1/ZT3,  2—  1/H3  ?         4n«.  r6—  4*4+  22**—  25x—  42  =  0. 

6.  One  root  of  the  equation 

x«_5;r8-f  13x—  21  =  0 
is  -j-3  ;  what  is  the  reduced  equation  ?     Ans.  x*  —  2x-{-7  =  0. 

7.  One  root  of  the  -equation 

Z4-{-2x3—34.r2  4-12*4-35  rrr  0 

is  —  7  ;  what  is  the  depressed  equation  ? 

Ans.  x3  —  5x*+x-\-5  =  0. 

8.  Two  of  the  roots  of  the  equation 

x'—Sx*—  4*"+30a-—  36  =  0 

are  -f-2,  —  3  ;  what   is   the  depressed  equation,  and  what  are    its 
roots  ?  r          The  depressed  equation  is 

Ans.  J  x*  —  kt'4-6  =  0  ; 

(and  its  roots  are  2+l/~2,  2—1/^2. 

4t2O.  -4w,?/  equation  having  fractional  coefficients  can  be  trans- 
formed into  another  in  which  the  coefficients  are  entire,  that  of  the 
first  term  being  unity. 

If  the  coefficient  of  the  first  term  of  the  given  equation  is  not 
unity,  make  it  so  by  dividing  through  by  this  coefficient.  Then  the 
equation  will  be  of  the  form. 

xm+Axm-l-{-Bxmr*-\-  ____  Tx-\-  U  =  0, 

in  which  it  is  supposed  that  some  or  all  the  coefficients,  A,  B,  etc., 
are  fractional. 

Assume  x  =  —  ,  (/  being  entirely  arbitrary,  and  substitute  this 
a 

value  of  x  in  the  equation  j   it  then  becomes 

£_M*ri  +  j£:         .7-^+^7=0. 

am  ~        a"1-1  n        a"1-2  ~  a    ' 

Whence,  by  multiplying  through  by  aTO, 


'»-*+  ....    T(j»-*y+  Uam  =  0. 
Now  since  a  is  arbitrary,  its  value  may  be  so  selected  that  it  and  its 


368  PROPERTIES   OF   EQUATIONS. 

powers  will  contain  the  denominators  of  tlic  fractional  coefficients  of 
the  original  equations.  We  present  the  following  examples  for  illus- 
tration. 

1.  Transform  the  equation, 

x.+  -°  +  -  +  -  =  o, 

in         u     '  p 

into  another  which  shall  have  no  fractional  coefficients,  and  which 
shall  have  unity  for  its  first  coefficient. 

Make  x  =  — —  ;  substituting  this  value  of  a*,  the  equation  bc- 
ninj>  ' 

comes 

7/s  ml*  l>i/         c 

•'  I  J  \         •'          [     _  _._  Q 

?}i8y<3^3       flu3//*/?9      nih*p      p  ~ 
Multiplying  every  term  of  this  by  •msn3j>3 ,  we  havo 

y*-\-cuipy*-\-l>nilnp!ly-}-cm*>*pt  =.  0. 

When  the  denominators  of  the  coefficients  have  common  factors, 
we  may  make  x  equal  to  y  divided  by  the  least  common  multiple  of 
the  denominators. 

ax9       bx       c 

2.  Transform  the  equation  x  -] -J -I —  =  0,  into  another 

1   2>})l       m      p 

which  shall  have  no  fractional  coefficients,  and  that  of  the  first  term 
be  unity. 

To  effect  this  it  is  sufficient  to  put  x  =  —.    With  this  value  of 

pm 

x  the  equation  becomes 

jf__    ,     Wf_    ,   _^_  _,   1  _Q 
2>3iu*     jni?    '  pm*      p 

Multiplying  every  term  by^s??i3,  we  obtain 

f+ftf+lp^my^fm*  =  0 
for  the  transformed  equation  required. 

8.  Transform  the  equation  x*-\-  ~  -\-  ——  -f-  — •-  -[--—  =  0  into 

another  having  no  fractional  coefficients. 

AM.  ,y4-r-20/-|-18-24^-f-7(24)V+2(21)8  =  0. 


PROPERTIES    OF    EQUATIONS.  360 

In  transforming  an  equation  having  fractional,  into  another  with 
entire  coefficients,  in  terms  of  another  unknown  quantity,  it  is  im- 
portant to  have  the  transformed  equation  in  the  lowest  possible 
terms.  The  least  common  multiple  of  the  denominators  will  not 
necessarily  be  the  least  value  of  a  that  will  give  the  required  equa- 
tion. If,  in  each  case,  the  denominators  be  resolved  into  their 
prime  factors,  it  will  be  easy  to  decide  upon  the  powers  of  these 
factors  to  be  taken  as  the  factors  of  a. 

The  following  illustration  will  render  further  explanation  unnec- 
essary. 

4.  Transform  the  equation, 

3  13  17 

^.3  _       __  ~2       I  _    ™   _ 

"^ 


_ 

35       "2450         68600" 

into  another  of  the  same  form  with  the  smallest  possible  entire 
coefficients. 

Writing  y  for  x  and  multiplying  the  second,  third  and  fourth 
terms,  by  a,  a5,  a9,  respectively,  we  have 

o  I  q  I  "T 

y'  _  &-  ay+  __  a.x_  «»  =  0. 


The  denominators,  resolved  into  their  prime  factors,  are 

7-5,     7a-53-2,     73-5a-23; 

and  assuming  a  =  7  *  5  •  2,  the  equation  may  be  written 
3  -7  -5  -2         13  -r-tf'2*        17-73-53-23 


which  reduces  to 

y_6y'+26y—  85  =  0. 

In  this  example,  the  least  common  multiple  of  the  denominators  is 
78  •  52  •  28  }  and  had  this  value  been  taken  for  a,  instead  of  7*5-2, 
the  coefficients  of  the  transformed  equation  would  have  been  much 
larger  than  they  are,  as  found  above. 

When  a  root  of  the  transformed  equation  is  known,  the  corres- 
ponding root  of  the  original  equation  will  be  given  by  the  relation 


8TO  PROPERTIES  OP  EQUATIONS. 


COMMENSURABLE  ROOTS. 

4SO.  A  number  is  commensurable  with  unity  when  it  can  be 
expressed  by  an  exact  number  of  units  or  parts  of  a  unit;  a  num- 
ber which  can  not  be  so  expressed  is  incommensurable  with  unity. 

431.  Every  equation  having  unity  for  the  coefficient  of  the  fir^t 
term,  and  for  nil  the  other  coefficients,  whole  numbers,  can  have  only 
ichole  numbers  for  its  commensurable  roots. 

This  being  one  of  the  most  important  principles  in  the  theory 
of  equations,  its  enunciation  should  be  clearly  understood.  Such 
equations  may  have  other  roots  than  whole  numbers  ;  but  its  roots 
can  not  be  among  the  definite  and  irreducible  fractions,  such  as  ~,  J, 
•ip,  etc.  Its  other  roots  must  be  among  the  incommensurable 

quantities,  such  as  j/2,  (S)"5,  etc.;  i.  e.,  surds,  indeterminate  deci- 
mals, or  imaginary  quantities. 

To  prove  the  proposition,  let  us  suppose  —  ,  a  commensurable  but 

irreducible  fraction,  to  be  a  root  of  the  equation, 

xm-\-  Ax^+Bx™-*  ....  Tx+  U=Q, 
A,  B,  etc.,  being  whole  numbers. 

Substituting  this  supposed  value  of  x,  we  have 


Transpose  all  the  terms  but  the  first,  and  multiply  by  bm~ly  and 
we  have 


Now,  as  a  and  b  are  prime  to  each  other,  b  can  not  divide  a, 
or  any  number  of  times  that  a  may  be  taken  as  a  factor  ;  for 

—  being  irreducible,  —  X  &  is  also  irreducible,  as  the  multiplier  a  will 

not  be  measured  by  the  divisor  b  ;  therefore  —  can  not  be  expressed 

am 
in  whole  numbers.     Continuing  the  same  mode  of  reasoning,  -—  • 


COMMENSURABLE    ROOTS.  371 

can  not  express  a  whole  number,  but  every  term  in  the  other  mem- 
ber of  the  equation  expresses  a  whole  number. 

Hence,  the  supposition  that  the  irreducible  fraction     is  a  root  of 

the  equation,  leads  to  this  absurdity,  that  a  series  of  whole  numbi^s 
is  equal  to  an  irreducible  fraction. 

Therefore,  we  conclude  that  any  equation  corresponding  to  these 
conditions  can  not  have  a  definite  commensurable  fraction  among  its 
roots. 

43^2.  It  has  been  shown  (42O)  that  an  equation  having  fraction- 
al coefficients,  that  of  the  first  term  being  unity,  can  be  changed  in- 
to another  of  the  same  form,  with  entire  coefficients.  The  expres- 
sion entire  must  there  be  understood  in  its  algebraic  sense ;  that  is, 
the  new  coefficients  being  entire  merely  in  algebraic  form,  may  be 
irrational  or  imaginary.  In  the  preceding  article  it  is  proved  that 
if  these  coefficients  are  whole  numbers,  all  the  commensurable  roots 
of  the  equation  are  also  whole  numbers ';  moreover,  these  roots  must 
be  found  among  the  divisors  of  the  absolute  term  ;  (428).  If  the 
divisors  of  the  absolute  term  are  few  and  obvious,  those  answering 
to  the  roots  may  be  found  by  trial  substitutions ;  but  in  most  cases 
the  labor  will  be  abridged  by  the  rule  suggested  by  the  following 
investigation : 

Suppose  a  to  be  a  commensurable  root  of  the  equation, 

z"t-}-J.xm-14- -f^x3-f  Sj?+Tjc+U=  0. 

Writing  a  for  -ce,  transposing  'all  the  terms  except  the  last  to  the 
second  member,  and  dividing  through  by  a,  we  have 

-  =  —a"-1  -  Aam-*—. . . .  —Rcf—Sa—T. 
a 

But,  since  a  is  a  root  of  the  equation,  —  is  an  entire  number;  trans 

pose  — T^to  the  first  member  of  the  last  equation,  make \-  T  — 

d 

N} ,  and  divide  both   members  of  the  resulting   equation  by  a ;  it 
then  becomes 

N 

1  =  — am~z — Aa~-* — —Ra—S. 

a 

The  second  member  of  this  equation  is  a  whole  number;  the  first 


372  PROPERTIES    OF    EQUATIONS. 

member  is  therefore  entire  ;  and  if  —  S  be  transposed  to  this  mem- 
ber, and  -—  -f  S  be  denoted  by  .A7".,,  we  shall  again  have,  after  di- 
viding through  by  a,  the  equation, 


=  -</*-  3—  -1 
a 


of  which  the  second,  and  therefore  the  first  member,  is  an   entire 
number. 

By  continuing  this  process  of  transposition'  and  division,  we  slv.ll 
finally  arrive  at  the  equations, 


^i=_1)0r^:i+1  =  0. 
a  a 

Every  whole  number  which  is  a  root  of  the  proposed  equation 
will  satisfy  all  of  the  above  conditions,  from  the  first  down  to  that 
expressed  by  the  equation, 


by  which  the  root  will  be  recognized. 

All  of  the  commensurable  roots  of  an  equation  of  the   assumed 
form  may  then  be  found  by  the  following 

RULE. — I.    Write  all  of  the  exact  divisors  of  the  absolute  term  m 
a  line,  and  beneath  them  write  their  rcsjwcttve  quotients. 

II.  Add  to  these  quotients,  severally,  the  coefficient  of  the  next  to 
the  last  term,  with  its  proper  si;n. 

III.  Divide  such  of  these  sums  by  the  divisors  to  which  they  cor- 
respond as  will  f five  exact  quotients,  neglecting  others. 

IV.  Add  to  these  quotients  the  coefficient  of  fhe  third  term  from 
the  last,  icith  its  proper  siyn,  and  divide  again  as  before,  and  so  on, 
until  the  coefficient  of  the  second  term  has  been  added  to  the  prece- 
ding quotient?,  and  these  last  in  turn  are  divided  Ity  their  respective 
divisors.      Those  divisors  which  correspond  to  the  final  quotients, 
minus  1,  are  roots  of  the  equation. 

NOTE.— Absent  powers  of  the  unknown  quantity  must  be  introduced 
with  iO  for  tlicir  coefficient. 


COMMEXSURABLE    ROOTS.  373 

EXAMPLES. 

1.  Required  the  commensurable  roots  (if  any)  of  tlic  equation, 
<  s«-f  4*'— S^+Sx— 10  =  0. 

OPERATION. 

Divisors,         10,         5,         2,  1,       -  1,     —2,     —5,     —10. 

—10 


Quotients,  —  1, 
Add                 8 

_2,     -5,    -10, 

10, 

5,        2,          1. 

•7, 

2d  quotients, 
Add 

3d  quotients, 
Add 

4th  quotients, 

0,        3,       -2, 

o 

-) 

—3 

18, 
-18, 

13,       10,          9 
2. 

-5, 
-5, 
4 

-21, 
21, 

—5. 
1. 

-1, 
i 
*i 

25, 
-25, 

5. 
—1. 

Thus,  there  are  two  final  quotients  equal  to  —  1  ;  and  the  corres- 
ponding divisors  arc  1  and  —  5.  Hence,  the  given  equation  has  two 
commensurable  roots,  1  and  —  5. 

If  we  divide  the  given  equation  by  (x  —  l)(x-f-5),  or  x*-}-4.x  —  5, 
the  quotient  will  be  x3  —  2.  Hence,. 

x8—  2  =  0,  x=±1/2; 

and  the  four  roots  are  1,  —  5,  -f-|/2,  —  -j/2. 

2.  Required  the  commensurable  roots  of  the  equation,  x9  —  6x* 
+  1  la?-  6  =  0.  Ans.  1,2,3. 

3.  Required  the  commensurable  roots  of  the  equation,  x'+4x*  — 
#»—  16x—  12  =  0.  Ans.  2,  —1,  —2,  —3. 

4.  Required  the  commensurable  roots  of  the  equation,  x*  —  6x3  _ 
16*4-21=0.  Ans.  Sandl. 

NOTE.—  Supplying  the  absent  term,  the  equation  will  be  *4±0.^—  G^a 
—  lO.r-i-21  =  0.  In  the  operation,  go  through  the  form  of  adding  0. 

5.  It  is  required  to  find  all  the  roots  of  the  equation  x*  —  Gxs-j- 


5.4*+2x--10  =  0.  Ans.  -1,  -}  5,  1+  1/~1,  l- 


374  PROPERTIES    OF    EQUATIONS. 


DERIVED    POLYNOMIALS. 

433.  The  first  member  of  an  equation  involving  but  one 
unknown  quantity,  to  which  all  the  terms  have  been  transposed,  is 
a  polynomial  in  the  most  general  sense  of  the  term,  and  may  be 
operated  on  as  an  algebraic  quantity,  without  reference  to  the  equn 
tion  and  to  the  particular  values  of  the  unknown  quantity  which 
will  reduce  the  polynomial  to  zero,  or  satisfy  the  equation. 

If  we  take  the  polynomial, 

Axm-\-Bxm~l-\-  (7xm-3-f -|-£c2-f-  TV -f-  U, 

and  multiply  each  term  by  the  exponent  of  x  in  that  term,  then 
diminish  this  exponent  by  unity,  and  form  the  algebraic  sum  of  the 
results,  we  shall  have 

m^x—'+Cm— l)J&x"1-a4-(m— 2)  Cxm~3+ ....  +2Sx+  T. 

Constructing  a  third  polynomial  from  this,  in  the  same  way  that 
this  was  derived  from  the  first,  we  have  m(m — l).4xiW~2-f 

(m— l)(m— 2)JRrl"~8-f  (m— 2)(m— 3)C7.r"-4+ +2S. 

A  fourth  may  be  formed  from  the  third  according  to  the  same 
law ;  and  so  on,  until  we  arrive  at  an  expression  which  will  be 
independent  of  x,  because  the  degree,  with  respect  to  x,  of  any 
polynomial  thus  formed,  is  one  less  than  of  that  which  immediately 
precedes  it. 

Denoting  the  given  polynomial  by  X,  the  second  by  Xl,  the  third 
by  X.2,  etc.,  then 

Xl  is  the  first  derived  polynomial  of-JT, 
X2  «  «  "  JT,, 

X3  "  "  "  Xo,  etc. 

And  Xlf  JT2,  JTg  are  the  successive  derived -polynomials  of  ^T, 
and  are  called  first,  second,  third,  etc.,  derived  polynomials. 

Preserving  the  above  notation,  we  have  for  the  successive  derived 
polynomials,  the  following 

RULE. —  To  form  X,,  multiply  every  term,  of  X  by  the  exponent 
of  small  x  in  the  term,  then  diminish  this  exponent  by  unity  and  take 
the  algebraic  sum  of  the  results.  X2  is  derived  from  Xj  in  the  same 
way  that  X,  is  derived  from  X;  and  so  on. 


DERIVED    POLYNOMIALS.  375 

What  are  the  successive  derived  polynomials  of 
-9x3-j-7x2 — 8x-j-5  ? 

1st,   5  •  3x4-|-4  •  5x3— 3  •  9x2-f2  •  7x— 8; 
2d,    4-5-3x8-f3-4-§#;2— 2-3-9x-}-2-7; 
3d,    3-4-5-3x2+2-3-4-5x— 2-3-9; 
4th,  2-3-4-5-3x+2-3-4-5; 
k5th,  2  •  3  •  4  •  5  •  3. 


('• 
$ 


COMPOSITION   OF   DERIVED    POLYNOMIALS. 

434.  Let  us  take  the  polynomial, 

xm-f^^m-14-^xm-a-f -|-  Tx+  U  =  X-, 

and  suppose  that  its  binomial  factors  of  the  first  degree  with  respect 

to  x  are 

x — a,  x — bj  x — c, ....  jX — m,  x — n. 

We  shall  then  have  the  identical  equation, 

which  will  subsist  as  a  true  equation,  whatever  quantity  be  substituted 
for  x  in  its  two  members.     Replace  x  by  y-\-x ;  then 


in  which  the  terms  re  —  a,  cc  —  6,  may  be  regarded  as  single,  and 
hence  the  factors  of  the  second  member  as  binomial.  Now  the 
terms  of  the  first  member  of  this  equation,  developed  and  arranged 
with  reference  to  the  ascending  powers  of  y,  will  give 


And  if  the  second  member  be  developed,  and  arranged  in  the 
same  manner,  then  by  (4S4?  5),  the  coefficient  of  y°  will  be 
(x  —  CL)(X  —  &)  ....  (x  —  m)(x  —  n). 

The  coefficient  of  y  must  be  the  algebraic   sum   of  the  products 
of  the  factors  x  —  a,  x  —  6,  etc.,  taken  m  —  1  in  a  set. 

The  coefficient  of  y*  must  be  the  algebraic  sum  of  the  products 
of  these  factors  taken  m  —  2  in  a  set. 

In  short,  these  coefficients  may  all  be  formed  according  to  the  law 
which  governs  the  product  of  any  number  of  binomial  factors 


376  PROPERTIES  OF  EQUATIONS. 

But  the  coefficients  of  the  like  powers  of  cy  in  these  two  devel- 
opments must  be  equal •;  (368,  III).  Hence, 

X  =  (x — a)(x — l>}(x — c) ....  (x — ni)(x — n)  ; 
and  since  the  sum  of  all  the  products  that  can  be  formed  by  multi- 
plying m  factors  in  sets  of  m — 1  and  m — 1,  is  the  same  as  the  sum 
of  all  the  quotients  which  can  be  obtained  by  dividing  the  continued 
product  of  tho  factors,  by  each  factor  separately,  it  follows  that 

X  X  X          X 

A1=  -    -H -_{-...._}- j-- 

x — a       x — b  x — m       x — n 

So  likewise  the  sum  of  the  products  of  the  binomial  factors 
taken  m — 2  and  m — 2,  is  the  same  as  the  sum  of  all  the  quotients 
obtained  by  dividing  the  continued  product  by  all  the  different 
products  of  the  binomial  factors  taken  2  and  2  ;  that  is, 

JL'a X_ X_  X_ 

o      —    /  ._        _.  \  f  —        /  \      I      /  _..        „>  (  „.        ,.\      I     •  •  *  "     I 


By  like  reasoning  it  may  be  shown  that 

*. £ ,  ,    £ 

2-8  -   (x—a\x—V)(x-c)  1  t  (*-a)  (x-wi)  (x— *i 

x  A  so  for  the  next  coefficient  in  order,  etc.,  etc. 


EQUAL   ROOTS. 

43*3.  It  has  been  seen  (427)  that  if  a,  6,  c,. . .  .,m,  n  arc  the 

roots  of  the  equation, 

X  =  xm+ Axn~l+Bx'»-*+ ; . . .  +  3Tx+  U  =  0, 
it  may  be  written, 

X=  (x—a}(x—l~}(x—c]    .  .  .(x—m}(x—n)  —  0. 
Now  if  a  number  p  of  these  roots  are  each  equal  to  «,  a  number 
q  equal  to  b,  and  a  number  r  equal  to  c,  the  last  equation  becomes 

X—  (x—a}P(x—b^(x—c)r (x—m}(x—n}  =  0. 

But  since  A"  contains  p  factors  equal  to  x — a,  q  factors  equal  to 
x — i}  r  factors  equal  to  x — c,  its  first  derived  polynomial  will  con- 

X  X 

tain  the  term  p  times,  the  term    —^—.      a  times,  the  term 

•x.~~«   '  x — b 


EQUAL    ROOTS.-  377 

Y  Y          X 

— —     r  times,  besides  the  terms    >    ,    etc.,   corrcspond- 

a: — c  x — m     x — u 

ing  to  the  single  roots,  (434)  ;  that  is, 

„    Px  ,.^+j^+__  +_^.  +^_. 


The  factor  (x  —  a)p  is  found  in  every  term  of  this  expression  for 
A",  except  the  first,  from  which  one  of  the  p  equal  factors,  x  —  «, 
has  been  suppressed  by  division.  Hence,  (x  —  a}p~l  is  the  highest 
power  of  x  —  a,  which  is  a  factor  common  to  all  the  terms  of  A",. 

For  like  reasons  (x  —  &)*"1,  (x  —  c)r~*  are  the  highest  powers  of 
the  factors  x  —  &,  x  —  c,  which  arc  common  to  all  the  terms  of  X1  ; 
hence, 


is  the  greatest  common  divisor  which  exists  between  the  first  member 
of  the  proposed  equation  audits  first  derived  polynomial. 

The  supposition  that  the  given  equation  contains  one  or  more  sets 
or  species  of  equal  roots,  necessarily  leads  to  the  existence  of  this 
greatest  common  divisor.  Conversely  :—  if  there  be  a  common  di- 
visor between  A"  and  A",  there  must  be  one  or  more  sets  of  equal 
roots  belonging  to  the  equation. 

For,  if  (x  —  a)'  be  a  factor  of  the  greatest  common  divisor,  then 
the  composition  of  A",  shows  that  (x-  —  a)'*1  is  a  factor  of  A*,  and 
that  a  is  therefore  t-^-l  times  a  root  of  the  equation  A"=  0.  Hence 
the  conclusions  : 

1.  —  An  equation  involving  but  one  unknown  quantity,  x,  and  of 
which  the  second  member  is  zero,  has  equal  roots  ii  there  he  between 
its  first  member,  X,  and  its  first  derived  polynomial  A",,  a  common 
divisor  containing  x. 

2.  —  The  greatest  common  divisor,  D,  of  X  and  AT,  ,  is  the  product 
of  those  binomial  factors  of  Ar,  of  the  first  degree  with  respect  to  x, 
which  correspond  to  the  equal  roots,  each  raised  to  a  power  whose  ex- 
ponent is  one  less  than  that  with  which  it  enters  A".     Therefore, 

To  determine  whether  an  equation  has  equal  roots,  and  if  so,  to 
find  them,  if  possible,  we  have  the  following 

E.ULE.  —  I.  S'-ek  the  greatest  common  divisor  between  the  first 
member  of  the  proposed  equation  and  its  first  derived  polynomial. 
32* 


878  PROPERTIES  OF  EQUATIONS. 

If  no  common  divisor  be  found,  there  are  no  equal  roots;  but  if  one 
l>e  found,  there  are  equal  roots  ;  in  which  case 

II.  Make  an  equation  by  placing  the  greatest  common  devisor,  D, 
equal  to  zero  ;  then  any  quantity  which  is  once  a  root  of  D  =  0 
ic  ill  be  twice  a  root  of  X  =  0j  any  Quantity  which  is  twice  a 
root  of  D  =  0  will  be  three  times  the  root  of  X  —  0  ;  and  so  on. 

It  will  at  once  be  seen  that,  if  D  contains  a  factor  of  the  form 
(x  —  a)1,  t  being  a  positive  whole  number  greater  than  unity,  and 
we  denote  the  greatest  common  divisor  which  exists  between  D  and 
its  first  derived  polynomial  D11  by  Dr,  then  D'  will  contain  the 
factor  (x  —  a)*"1.  And,  again,  denoting  by  Z>',  the  first  derived 
polynomial  of  Z>',  and  by  D"  their  greatest  common  divisor, 
(x  —  a)'~2  will  be  a  factor  of  D".  This  process  being  continued,  as 
the  exponent  of  (x  —  a),  —  and  consequently,  the  degree  of  the 
greatest  common  divisor,  —  diminishes  by  one  for  each  operation,  it 
is  plain  that  when  the  degree  of  the  equation, 

#=;i>; 

is  too  high  to  be  solved,  we  may  in  certain  cases  make  the  determin- 
ation of  the   equal  roots  depend  upon  the  solution  of  equations  of 
lower  degrees,  until  finally  one  is  obtained  which  can  be  solved. 
To  illustrate,  suppose  that  for  the  equation, 

JT=0, 
it  is  found  that 

D  =  (x—  a)\x—  b)n(x—  c)  ; 
then  D'  =  (x—  a)"-i(a>-  &)*-», 

D"  =  (x—  ay-*(x—  &)*-», 


The  equation, 

/*—  «>  =  (x—  a)(x—  6)  =  0, 

may  be  solved,  giving  the  roots  x  =  a,  x  =  b,  and 
(*-a)*M,  (*-&)«+',  (x—  c)', 

are  factors  of  X,  or  a  and  b  are  each  n-}-l  times  roots,  and  c  twice  a 
root,  of  the  equation, 

X=0. 
Dividing  the  given  equation  by  the  product, 

(x—  «)"+  »  (x—  6)"+  »  (x—  c)3, 
its  degree  will  be  depressed  2n-{-4  units. 


EQUAL   KOOTS.  379 


EXAMPLES. 

1.  Docs  the  equation  x4  —  2x8  —  7xa-j-20x  —  12  =  0,  contain  equal 
roots,  and  if  so,  what  are  they  ? 

The  first  derived  polynomial  of  the  first  member  is 

4x3—  6x2—  14.T+20. 

The  greatest  common  divisor  between  this  and  the  first  member  of 
this  equation  is  x  —  2  ;  therefore  x  =  2  is  twice  a  root  of  the  equa- 
tion, and 

x*—2x3—  7z'-f20.r—  12 

n:ay  be  divided  twice  by  x  —  2,  or  once  by  (x  —  2)2  =  x9  —  4./--J-4. 
Performing  the  division,  we  find  the  quotient  to  be  xa-j-2x  —  3,  and 
the  original  equation  may  now  be  written 

(xs—  4x-|-4)(a;a+2a?—  3)  =  0. 

This  equation  will  be  satisfied  by  the  values  of  x  found  by  placing 
each  of  these  factors  equal  to  zero.  From  the  first  we  get  x  =  2, 
x  =  2,  and  from  the  second  x  =  1  x  =  —  3  ;  hence  the  four  roots 
of  the  given  equation  are  1  ,  2,  2,  —  3. 

2.  Find  the  equal  roots  of  the  equation 

x'+^-llx*—  8x'+20x+16==0.    Ans. 

3.  What  arc  the  equal  roots  of  the  equation 

x»_2x4-f3^8—  7xa-f-8x—  3  =  0? 

Ans.  It  has  three  roots,  each  equal  to  1. 

4.  What  are  the  roots  of  the  equation 

x*—2x*—  lla?+  12o:+36  =  0  ? 

Ans.  Its  roots  are  ,3,  3,  —2,  —2. 
5    What  are  the  equal  roots  of  the  equation 

X=  x1—  5o;8—  2z5-f-3Sx4—  31x8—  61#'-f96;c—  36  =  0? 
We  find 


D  =  x*—  3*8—  x'-j-llz—  0, 


Df  =  x—l. 

Hence  or-  =  1  is  twice  a  root  of  the  equation  D  =  0,  and  three  times 
a  root  of  the  given  equation. 


880 


PKOPERTIES    OF    EQUATIONS. 


Dividing  D  —  x'—  3xz—  x*+IIx—  6   by   Z)'2  =  x*—  2x+ 
fiad  for  the  quotient  x2—  x—  6  =  (x—  3)(x-r2).     Therefore, 


and  X=  (x—  3)2(x-f2)20r—  1)». 

Hence  the  roots  of  the  given  equation  are 

Ans.  3,  3,  -2,  -2,  +1,  +1,  +1. 

436.  Having  an  equation  involving  Lut  one  unknoicn  quantity, 
to  transform  it  into  another,  the  roots  of  which  shall  differ  from 
those,  of  the  proposed  equation  l)y  a  constant  quantity. 

Assume      xm-\-Axm-l-\-£xm-*+  Cxm-3+  ____  -j-  TJC+  U=Q, 
and  denote  the  new  unknown  quantity  by  y,  and  by  x'  the  arbitrary 
but  fixed  difference  which  is  to  exist  between   the  corresponding 
values  of  x  and  y\  we  shall  then  have  x  =  y-\-x'. 

Substituting  this  value  of  x  in  the  given  equation,  it  becomes 


Developing  the  terms  separately,  by  the  binomial  formula,  and 
ranging  the  aggregate  o 
ing  powors  of  y,  we  have 


arranging  the  aggregate  of  the  results  with  reference  to  the  ascend- 


+ TX' 


y/-j-  nix 

-f  .4(??i—  IX"1-2 


C(m—  3X 


m-l 


— 2)  --  x'—' 
2 


m—l 


xn 


=  0.       (1) 


An  examination  of  this  developed  first  mciubcr  leads  to  these 
conclusions  : 

1. — The  absolute  term  of  the  transformed  equation,  or  the  coeffi- 
cient of  y,  is  what  the  first  member  of  the  given  equation  becomes 
when  x'  is  substituted  for  x. 


TRANSFORMATIONS.  381 

3.  —  The  coefficient  of  y,  the  first  power  of  the  unknown  quantity, 
is  what  the  first  derived  polynomial  of  the  first  member  of  the  given 
equation  becomes,  when  in  it  x'  takes  the  place  of  x. 

3.  —  The  coefficient  of'  y*  is  what  the  second  .derived  polynomial 
of  the  first  member  of  the  given  equation  becomes  when  it  is  divided 
by  2,  and  x'  takes  the  place  of  x. 

4-  —  And  in  general,  the  coefficient  of  yn  is  what  the  nth  derived 
polynomial  of  the  first  member  of  the  given  equation  becomes  when 
it  is  divided  by  the  product  of  the  natural  numbers  from  1  to  n  in- 
clusive, and  x  is  replaced  by  x'. 

Representing  the  first  member  of  the  given  equation,  and  its  suc- 
cessive derived  polynomials,  after  x'  has  been  substituted  for  x,  by 
A7,  A'7,,  A*'3,  A'3,  etc.  respectively,  the  transformed  equation  may 
be  written 


Or,  by  inverting  the  order  of  terms, 


437.  By  comparing  eqs.  (1)  and  (!')  of  the  preceding  article  it 
is  shown  that, 


•jrt  _  -i 

ttDd      1-2  .  .  (m-5)  =  m  ~2~  *"+^0»-lX+*  ;      (3) 
the  degree  of  the   coefficients  of  the  equation,  with  respect  to  cc', 
increasing  by  at  least  one  from  term  to  term  as  we  pass  from  left  to 
right,  the  absolute  term  being  of  the  Twth  degree. 

Now,  since  x'  is  an  arbitrary  quantity,  such  a  value  may  be  as- 
sumed for  it  as  will  cause  it  to  satisfy  any  reasonable  condition.  We 
may  therefore  form  an  equation,  by  placing  any  one  of  these  coeffi- 
cients equal  to  zero,  regarding  ./'  as  the  unknown  quantity,  and  any 
root  of  this  equation  will  cause  the  corresponding  term  of  the 
transformed  equation  (I'),  (43G),  to  disappear. 


882  PEOPERTIES    OF    EQUATIONS. 

Suppose 

mx'-\-  A  =  0  ;  -whence  x'  =  --- 

m 

If  this  value  of  x'  be  substituted  in  the  equation  just  referred  to,  it 
takes  the  form 


Hence,  to  transform  an  equation  into  another  which  shall  be  in- 
complete in  respect  to  the  second  term  :  Substitute  for  the  unknown 
quantity  another,  minus  the  coefficient  of  the  second  term  divided  by 
the  exponent  of  the  degree  of  the  equation. 

438.  The  third  term  will  disappear  from  the  transformed  equa- 
tion when  x'  is  made  equal  to  either  of  the  roots  of  the  equation, 

m  _  1 
m  ^—±  x'*+  A(m—  l)x'+  JB  =  0. 

2 

But  there  may  exist  such  a  relation  between  m,  A,  and  B,  that  the 

A 

value,  :c'  =  --  ,  will  satisfy  this  equation  :  in  which  case  the  van- 
m 

ishing  of  the  second  term  of  the  transformed  equation  will  involve 
that  of  the  third.  To  find  what  this  relation  is,  substitute  this 
value  of  x'  in  the  above  equation,  and  it  becomes 


2         ma       v         J  m 
This,  reduced  as  follows, 

^Lz.1.  £'_(m_i)£+jB=o, 

2         m       v          '  m 
(Wl-_l)^[9_2(m— 1)  J.'-f  2m.g  =  0, 
(m— 1)^L2  = 


gives  finally  = 

When  the  values  m,  A,  and  B,  will  satisfy  this  equation,  the  third 
term  of  the  transformed  equation  will  disappear  with  the  second. 
In  general,  to  find  the  value  of  xf  which  will  free  the  transformed 
equation  of  the  third  term,  an  equation  of  the  second  degree  must 
be  solved ;  and  to  free  it  of  the  fourth  term,  the  equation*  to  be 
solved  would  be  of  the  third  degree;  and  finally,  to  make  the  abso- 
lute term  disappear,  would  require  the  solution  of  the  original  equation. 


TRANSFORMATIONS.  383 


EXAMPLES. 

1.  Transform  the  equation  x*  -\-2px  —  q  =  0,  into  another  which 

which  shall  not  contain  the  second  term. 

2» 
This  is  done  by  making  x  =  y  --  ~  =  y—  p         (437)  ; 

Lt 

whence,  by  (436), 

*'  =  00  f—  2x^-2 

X'  1=-;2(Jp)-2p  =  0; 


Therefore,  the  required  equation  13 

=  0, 


from  which  we  find  y  =  ±V  q-\-p*  ;    and   since   cc  =  y  —  -p,  the 
values  of  x  are  given  by  the  formula, 

x  =  — 


the  same  as  that  found  by  the  rule  for  quadratics. 

2.  Transform   the  equation    x*-±-px?  -\-qx-\-r  =  0,  into  one  not 
having  the  second  term. 


Make  x  =  y  —  ^-  ;  then 

*  -(-fM-fM- 


_i 

2-3  ~  2-3 

Hence,  the  equation  sought  is 


or,  bj  making  jl  —  gr  =  m,  and  9:  —  ^|  +r  =  n, 


384  PROPERTIES  OF  EQUATIONS. 

3.  Transform  the  equation 

x4—  12a;3-f-17x9—  Qx+  7  =  0, 

into  another  which  shall  not  contain  the  3d  power  of  the  unknown 
quantity. 

1° 

By  (437),  put  x  =  y+  —  ;  or  x  ==  B+y. 

Here  xr  =  3  and  m  =  4. 

X'    =  (3)4—  12(3)'-{-17(3)'-9(3)+7,or        A7    =-110. 

A7j  =  4(3)'—  3G(3)a-f-34(3)—  9,  or        A",  =  —123. 

4^  =  6(3)«-SO(3)+17,  or       £»  =  -  37. 


Therefore  the  transformed  equations  must  be 
y4—  37ya—  123y—  110  =  0. 
4.  Transform  the  equation 

a;3—  6x'-J-13x—  12  =  0, 
into  another  wanting  its  second  term. 
x  =  2-fy;    then 

=  (2)'—  G(2)3-}-13(2)—  12,  or        A'7    =—2. 

=  3(2)'—  12(2)+13,  or        A",  =  +1. 

or         £?-=       0. 


Therefore,  the  transformed  equation  must  bo 

-2  =  0. 


5.  Transform  the  equation 

x*__4.x»_8x+32  =  0, 
into  another  whose  roots  shall  be  less  by  2. 

Put  x  -  2+y.  Ans.  y4+4y*—24y  =  0. 

As  this  transformed  equation  has  no  term  independent  of  y.t 
y  —  0  is  one  of  its  roots,  and  x  =  2  is  therefore  a  root  of  the 
original  equation. 


TllANSFOBMATIOXS,  385 

6.  Transform  the  equation, 


=  0, 
into  another  whose  roots  shall  be  greater  by  3. 

Ans.  y-f-4/4-9,y9—  42y  =  0. 

7.  Transform  the  equation, 

x«_Sx3-f  z'+82x—  60  =  0, 
into  one  incomplete  in  respect  to  its  second  term. 

Ans.  #*—  23,y8-f  22y+69  =  0. 
439.  Resuming  the  transformed  equation  (!'),  (436),  which  is 

=  0, 


and  replacing  y  by  its  value,  y  =  x  —  x',  it  becomes 


Now  it  is  evident  that,  by  developing  the  first  member  of  this 
equation  and  arranging  the  result  with  reference  to  the  descending 
powers  of  .r,  the  first  member  of  the  original  equation  will  be  repro- 
duced }  for,  by  this  operation  we  will  have  merely  retraced  the  steps 
by  which  eq.  (10  was  derived  from  eq.  (1)  in  the  article  referred  to 
Hence  we  have  the  identical  equation, 

±....Sx*+fx+U 


The  quotients  and  remainders  obtained  by  the  division  of  the  first 
member  of  this  equation  by  any  quantity,  will  not  differ  from  those 
arising  from  the  division  of  the  second  member  by  the  same  quan- 
tity. Dividing  the  second  member  by  x  —  xr,  the  first  remainder  is 
A'',  and  the  quotient, 


(*-*')'+ 


386-  PROPERTIES    OF    EQUATIONS. 

and  this  divided  again  by  x — xf,  will  give  for  the  second  remainder 
JTj  and  the  quotient, 

X9  X7  X9 

It  is  unnecessary  to  continue  this  process  further,  to  see  that 
these  successive  remainders  are  the  coefficients  of  the  transformed 
equation  (!')  beginning  with  the  absolute  term,  or  the  coefficient  of 
//.  The  divisor  to  be  employed  is  x — x'  if  the  roots  of  the  trans- 
formed equation  are  to  be  less,  in  value,  than  those  of  the  given 
equation  by  the  constant  difference  x'  j  if  greater,  the  divisor  must 
be  .r-j-.r.  Hence,  an  equation  may  be  transformed  into  another  of 
which  the  roots  are  greater,  or  less,  than  of  the  given  equation  by 
by  the  following 

RULE.  I.  Divide  the  first  member  of  the  given  equation  (the  sec- 
ond member  being  zero}  by  x  plus  the  constant  difference  between  the 
roots  of  the  two  equations,  continuing  the  operation  until  a  remain- 
der is  obtained  which  is  independent  of  x;  then  divide  the  quotient 
of  this  division  by  the  same  divisor,  and  so  on,  until  m  divisions  have 
been  performed. 

II.  Write  the  transformed  equation,  making  these  successive  re- 
mainders the  coefficients  of  the  different  powers  of  the  unknown 
quantity,  beginning  with  the  zero  power. 

It  must  be  borne  in  mind  that  the  term  plus  in  this  rule  is  used 
in  its  algebraic  sense. 

By  a  little  reflection,  it  will  seem  that  the  mth  quotient  will  be 
the  coefficient  of  xm  in  the  original  equation,  and  that  this  will  also 
be  the  coefficient  of  the  highest  power  of  the  unknown  quantity  in 
the  transformed  equation. 

EXAMPLES. 

1.  Transform  the  equation, 

x4— 4x9— 8;e-h32  =  t), 

into  another  of  which  the  roots  shall  be  less  by  2. 
This  is  example  5  of  the  last  article.     Make 
x  =  2-j-^,  or  y  =  x  —  2  ; 


TRANSFORMATIONS.  387 


then  the  operation  is  as  follows : 

-2)*4— 4x3 — 8*4-32(*'— 2*a — 4*— 16 
x'—2x> 


— 2*'— 8* 
_-2*»+4*a 


—  4*3_8*  a;—  2)*'—  2*a—  4*—  16(V—  4 

—  4*a4-8*  a;'--  2*a 


—16.^+32  —  4x—  16 

—  16jc-f32  —4*+  8 


•2X—  4(* 


__ 
—  2.r—  4  2-3 


Hence,  the  transformed  equation  is 

y4+4y±  0^—24^4-0=0; 
or,  y*+4y* — 24y  =  0,  as  before. 

2.  Transform  the  equation,  x*— 12x8+16-c9— 9*4-7  =  0,into  one 
having  roots  less  by  3. 

Here  *  =  y4~3>  ory  =  x — 3. 


OPERATION. 

>_9X_|_7(V_9*«— 10*— 39 
x4— 3*' 


— l(Kra-f30.;c 


—39.1-4-     7 
—39*4-117 


—  110  =  A17,     1st  remainder. 


PROPERTIES    OF   EQUATIONS. 
28 


*—  3xa 


—28*— 39 


—  123  = 


,  ,     2d  remainder. 

a—  3)*—  3(1 
a—  3 


—  28 
9 


«^       •»•  a 

—  37  =  -~>     3d  remainder. 

Hence,  #*±:0y  —  37,/  —  123^  —  110  =  0,  is  'the  transformed 
equation. 

We  shall  have  4  remainders,  if  we  operate  on  an  equation  of  the 
4th  degree  ;  5  remainders  with  an  equation  of  the  5th  degree  ;  andf 
in  general,  n  remainders  with  an  equation  of  the  nth  degree. 

The  transformation  of  equations  by  division,  treated  of  in  this 
article,  if  performed  by  the  ordinary  rule,  would  be  too  laborious  for 
practical  application  ;  but  by  a  modified  method  of  division,  called 
Synthetic  Division,  it  becomes  expeditious  and  easy. 

As  preliminary  to  the  explanation  of  this  method  of  division,  we 
must  explain  the  process  of 


MULTIPLICATION  AND  DIVISION  BY  DETACHED 

COEFFICIENTS. 

* 

4L4O.  It  has  been  seen  that  when  two  polynomials  are  homoge- 
neous their  product  is  also  homogeneous,  and  the  number  which  de- 
notes its  degree  is  the  sum  of  the'numbers  denoting  the  degrees  of 
the  factors.  It  is  evident  that  if  the  polynomials  contain  but  two 
letters,  and  both  are  arranged  with  reference  to  the  same  letter,  the 
product  will  be  arranged  with  reference  to  that  letter.  Since,  in 
the  operation  of  multiplying  the  terms  of  the  multiplicand  by  the 
terms  of  the  multiplier,  the  products  of  the  coefficients  are  not 


DETACHED    COEFFICIENTS.  889 

affected  by  the  literal  parts  to  which  they  are  prefixed,  these  coeffi- 
cients may  be  detached  and  written  down  with  their  signs  in  their 
proper  order,  and  the  multiplication  performed  as  with  polynomials. 
The  partial  products,  numerical  or  literal,  being  carefully  arranged 
as  if  undetached,  are  then  reduced  and  the  literal  parts  annexed. 


EXAMPLES. 

1.  Multiply  a'+2ax-j-o:2  by  a+x. 

OPERATION. 

l-j-2-j-l,  Detached  coefficients  of  multiplicand. 

1-|-1  "  "  multiplier. 

1+2+7 
1+2+1 

1+3+3  +  1  Product  of  coefficients. 

Now  by  annexing  the  proper  literal  parts  to  the  several v  terms 
thus  obtained,  we  have 

a3+3aax+3a;ra+z8;     Ans. 

This  method  of  multiplication  may  be  employed  when  the  two 
polynomials  contain  but  one  letter. 

2.  Multiply  3xa— 2;c— 1  by 


OPERATION. 

3 — 2 1 

3+2 

9— G— 3 
+6—4—2 

9+0— 7— 2 
whence, 

or,  9x3 — 1x — 2,     Ans. 

When  any  of  the  powers  of  the  letters,  between  the  highest  ana 
lowest,  do  not  appear  in   either  factor,  the  terms  corresponding  to 
such  powers  must  be  supplied,  with  the  coefficient  0. 
33* 


390  PROPERTIES  OF  EQUATIONS. 

3.  Multiply  x*+2x>— I  by  aa-f2. 

The   factors   completed  are    x'-\-2x*+Qx — 1   and   cc'-f-Oz-f  2. 

Hence  the  operation  i.s 

l-l-2-j-O—l 


1+2+0-1 

2+4+0-2 

1+2+2+3+0—2 
and  the  product, 

or,  ^+2x4+2x3+3x3— 2. 

4.  Multiply  3xa— 2x— 1  by  4x+2.    Ans.  12xs— 2xa— 8x— 2. 

5.  Multiply  3xa — 5x — 10  by  2x — 1.        Ans.  Qx3  —  22.c3 

6.  Multiply  o^+iry+y  by  xa — xy-\-y*.       Ans.  x4+xa 

7.  Multiply  xs— 4xa+5x— 2  by  x'+4x— 3. 

Ans.  x6— 14x*+30xa — 23x+6. 

441.  Now,  if  detached  coefficients  can  be  used  in  multiplication, 
so  in  like  cases,  they  may  be  employed  for  division.  When  the  divi- 
dend and  divisor  contain  but  two  letters  and  are  homogeneous,  the 
degree  of  the  quotient  will  be  the  excess  of  the  degree  of  the  div- 
idend over  that  of  the  divisor. 

EXAMPLES. 

1.  Divide  a4— 3a'z— 8aV+18ax'+16x4  by  a'— 2ax—2x\ 

OPERATION. 

1_3_8+18+16|1— 2— 2 
1—2—2  1—1—8 


__!_G-fl8-fl6 
-1+2+  2 


_8-fl6-j-lG 
Hence  the  quotient  is 


DETACHED    COEFFICIENTS. 


391 


2.  Divide  a6—  5a8Z>8+aV+6a£4—  W  by  a1—  Sa&'+JA 
In  this  example  we  must  supply  the  term  Oa*&  in  the  dividend, 
and  the  term  Q-a?l>  in  the  divisor.    The  operation  then  is, 
fl+6—  2 


l+Q—3  -fl  _       1+0— 

0—2+0+6—2 
_2+Q+6—  2 

Therefore  we  have,  for  the  quotient, 


or,  a— 

3.  Divide  x>—  4x4—  17a,3—  13xa—  llz—  10  by  cc'+3 

OPERATION. 

1—4—  17—  13—  II—  10  |l+3+2 
1+3+  2  __  i_7+2—  5 

_7_19_13-11—  10 
—  7—  21—  14 

+  2+  1—11—10 

+  2+6+4 


Hence  the  quotient  is 

When  the  dividend  and  divisor  contain  but  a  single  letter,  absent 
terms  in  either,  answering  to  powers  of  this  letter  between  the 
highest  and  lowest,  must  be  inserted  with  the  coefficient  0. 

In  the  examples  we  have  wrought  to  illustrate  the  method  of 
division  by  detached  coefficients,  the  coefficients  have  been  taken 
entire,  that  of  the  first  term  of  the  divisor,  in  each  case,  being 
unity ;  the  process,  however,  will  be  the  same  whatever  these 
coefficients  may  be.  When  the  coefficient  of  the  first  term  of  the 
divisor  is  not  unity,  it  may  be  made  so  by  dividing  both  dividend 
and  divisor  by  this  coefficient.  The  quotient  term  will  then  be  the 
first  term  of  the  corresponding  dividend,  as  is  seen  in  all  the  abova 
examples. 


392  PROPERTIES    OF    EQUATIONS. 


SYNTHETIC    DIVISION. 

.  To  explain  what  synthetic  division  is,  and  to  deduce  a 
rule  for  executing  it,  let  us  take  the  first  example  in  the  preceding 
article.  If  the  signs  of  the  second  and  third  terras  of  the  divisor 
be  changed,  each  remainder  will  be.  found,  by  adding  the  terms  of 
the  product  of  these  two  terms  by  the  term  of  the  quotient,  to  the 
corresponding  terms  of  the  dividend ;  observing  that  by  the  nature 
of  the  operation,  the  product  of  the  first  term  of  the  divisor  by  the 
term  of  the  quotie-nt,  cancels  the  first  term  of  the  dividend.  Be- 
sides, since  the  first  term  of  the  divisor  is  unity,  any  quotient  term 
is  the  same  as  the  first  term  of  the  partial  dividend  to  which  it 
belongs. 

The  process  may  now  be  indicated  as  follows : 


2—2—16 
2—  2—16 

Quotient,       1—1—8          0          0 

Hence  the  quotient  is  a2 — ax — 8x',  as  before  found. 

The  dividend  and  divisor  are  written  in  the  usual  way,  after 
changing  the  signs  of  the  last  two  terms  of  the  latter;  and  a  hori- 
zontal line  is  drawn  far  enough  beneath  the  dividend  for  two  inter- 
vening rows  of  figures.  Bring  down  the  first  term  of  the  dividend 
for  the  first  term  of  the  quotient.  The  products  of  the  second  and 
third  terms  of  the  divisor  by  the  first  term  of  the  quotient  are 
written,  the  first  in  the  first  row  under  the  second  term  of  the  div- 
idend, and  the  second  in  the  second  row  under  the  third  term  of 
the  dividend.  The  sum  of  the  second  vertical  column  is  ther 
written  for  the  second  term  of  the  quotient.  The  next  step  is 
multiply  the  second  and  third  terms  of  the  divisor  by  the  second 
term  of  the  quotient,  placing  the  first  product  in  the  first  row  under 
the  third  term  of  the  dividend,  and  the  second  in  the  second  row 
under  the  fourth  term  of  the  dividend.  The  sum  of  the  third 
vertical  column  is  the  third  term  of  the  quotient.  The  sums  of  the 
fourth  and  fifth  columns  each  reduce  to  zero. 

The  operation  for  the  last  example  in  the  preceding  article  is 


SYNTHETIC    DIVISION.  393 

1—4— 17— 13— -11—1011— 3— 2 
_3_|_21—  64-15 

_  2+14—  4+10 
1—7-f-  2—  5      0      0 
and  for  the  quotient  we  have 

x»_7x»_|_2x— 5. 

No  difficulty  will  now  be  experienced  in  understanding  this 
general 

RULE. — I.  If  the  coefficient  of  the  first  term  of  the  arranged 
divisor  is  not  unity,  make  it  so  by  dividing  loth  dividend  and  divisor 
by  this  coefficient. 

II.  Write  down  the  detached  coefficients  of  the  dividend  and  di- 
visor in  the  usual  way,  changing  the  signs  of  all  the  terms  of  the 
of  the   latter   except  the  first,  and  draw  a  line  far  enough  lelow 
the  dividend  for  as  many  intervening  rows   of  figures   as  there  ore 
terms,  less  one,  in  the  divisor,  and  bring  down  the  first  term  of  the 
dividend,  regarded  as  forming  a  vertical  column,  for  the  first  term 
of  the  quotient. 

III.  Write  the  products  of  the  second,  third,  etc.,  terms  of  the  di- 
visor by  the  first  term  of  the  quotient,  beneath  the  second,  third,  etc., 
terms  of  the  dividend  in  their  order,  and  in  the  first,  second,  etc.,  rows 
of  figures ;  and  bring    down  the  sum  of  the  second  vertical  column 
for  the  second  term  of  the  quotient. 

IY.  Multiply  the  terms  of  the  divisor,  exclusive  of  the  first,  as 
before,  by  the  second  term  of  the  quotient,  and  write  the  products  in 
their  respective  rows,  beneath  the  terms  of  the  dividend  beginning  at 
the  third  j  bring  down  the  sum  of  the  third  vertical  column  for  the 
third  term  of  the  quotient. 

V.  Continue  this  process  until  a  vertical  column  is  found  of  which 
the  sum  is  zero,  the  sums  of  all  the  following  aho  being  zero  when 
the  division  is  exact ;  otherwise  continue  the  operation  until  the  de- 
sired degree  of  approximation  is  attained.  Having  thus  found  the 
coefficients  of  the  quotient,  annex  to  them  the  proper  literal  parts. 

In  applying  this  method  of  division  it  is  unnecessary  to  write  the 
first  term  of  fhe  divisor,  since  it  is  unity  and  is  not  used  in  the  oper. 
ation. 

35 


394  PKOPERTIES    OF    EQUATIONS. 

EXAMPLES. 

1.  Divide  1— x  by  1+x.  Ans.  1— 2x+2za — 2xs+ctc. 

2.  Divide  1  by  1+z.  Ans.  1— x-f xa— x-'+z4— etc. 

3.  Divide  a6— 5a4x+10aV— lOaV-f  5ax4— x6  by  a8— 2ax+x*, 

Ans.  a3— 3aaz-|-3ax2— z3. 

4.  Divide  x9— bx'+lbx'— 24x«-j-27xa— 13x+5   by  *4— 2x3+ 
4xa— 2-r-f-l.  J.?is.  a;3— 3x--f5. 

5.  Divide  #T — yT  by  x — y. 

Ans.  x*+tfy+xyj&f+a*y*^?y*+?> 

4 4- 3.  The  transformation  of  an  equation  into  another  having 
roots  less  or  greater  than  those  of  the  given  equation  by  a  fixed 
quantity,  may  now  be  expeditiously  made  by  the  method  of  synthetic 

division. 

i .  Transform  the  equation  x* — 4x8 — 8^-|-^^  =  0»  in^°  another 
whose  roots  shall  be  less  by  two. 

The  second  power  of  x  not  appearing  in  this  equation,  it  must  bo 
introduced  with  ±0  for  its  coefficient. 

FIRST  OPERATION. 

1_4±0— 8+32j2_ 
2—4—8—32' 


1—2—4—16,     0  = 

SECOND    OPERATION. 

1_2 — 4— 16|2 
2-M)—  8'" 


lH-0—4,— 24  =  X\ 

THIRD    OPERATION.  FOURTH   OPERATION. 

1-4-0— 4|2  1+212 

2+4"  2~ 

1_L9       0  —  — ^  1     J-4  —  —* 

2  ~  2-3" 

Bence  the  transformed  equation  is 

— 24    =  0. 


SYNTHETIC    DIVISION.  »W> 

Instead  of  keeping  the  above  operations  separated,  they  may  be 
united  and  arranged  as  follows  : 

1_4  -4-0       -  8+32|2_ 
2  —4     —  8—32 

—2  —4    —16,    0  =  X 
2       0—8 


0  —4,    —24  =  ^', 
2_+4 

V7 

2,     -0=^ 

2_ 

i-5* 

"  2-3 

To  understand  this,  it  is  only  to  be  borne  in  mind  that  the  divisor 
is  the  same  throughout,  and  that  the  first  term,  1,  of  the  successive 
dividends,  \vhich  if  written  would  all  fall  in  the  vertical  column  at 
the  left,  is  omitted. 

Transform  the  equation  x* — 12x8-f-17xa — 9x-|-7  =  0,  into  an- 
other whose  root  shall  be  3  less. 

OPERATION. 

1  _12  -f  17  -       9  -|-     7    (3 
_j_  3  _27  _-  30  —117 
_  9  _io  _  39^—110  =  X* 
4.  3  —18  —  84 

_  6  —28,— 123=^7, 
4-  3  --  9 

o       07       -^a 
-  3,  -37  =  -g 

±J 

0  =  ^3 
2-3 

Hence  the  transformed  equation  -is 

y*+<y— 37^—123^—110  =  0. 

Transform  the  equation  x* — I2x — 28~?0,  into  another  whose 
roots  shall  be  4  less.  Make  x  =ty-f-4. 


396  PROPERTIES    OF    EQUATIONS. 

OPERATION. 
1      0      —12      —28      (4 

•4 


4  4 

4  32 

8  ""36  = 
4 


Hence  the  transformed  equation  must  bo 


Transform  the  equation  x3 — 10x3-J-3x — 6946  =  0,  into  another 
whose  roots  shall  be  less  by  20.  We  make  y  =  20-j-y. 

OPERATION. 

1     __10        3     —6946    (20 
_20    200        4060 

10     203     —2886 
20     600    ' 

"SO  ~803 

20    = 


The  three  remainders  are  the  numbers  just  above  the  double  lines, 
which  give  the  following  transformed  equation : 

y_|_50y-|-803y — 2886  =  0. 
Transform  this  equation  into  another  whose  roots  shall  be  less  by 

1     50      803     —2886    (3 
_3       159     -f-2886 

53    ~962         0 
3       168    ===== 


56     1130 
59 


PROPERTIES    OF    EQUATIONS.  397 

Hence  the  transformed  equation  is 


=  0. 
This  equation  may  be  verified  by  making  z  =  0 ;  which  gives 

y  —  3,  and  x  =  20+3=23. 

444.  If  the  signs  of  the  alternate  terms  of  any  complete  equa- 
tion involving  but  one  unknown  quantity  be  changedj  the  signs  of  all 
the  roots  will  be  changed. 
In  the  general  equation 

xm+Axm-l+ Bxm~*-\- +  Tx+  U=  0,      (1) 

let  the  signs  follow  each  other  in  any  order  whatever.  Changing 
the  signs  of  the  alternate  terms  of  this  equation,  beginning  with  the 
second,  we  have 

but  if  the  change  begin  with  the  first  term,  we  have 

— x^+Ax"-*  —  Bxm~~+ q=  Tx±  U=  0          (3) 

Now,  if  a  be  a  root  of  equation  (1),  its  first  member  reduces  to 
zero  when  a  is  substituted  for  x ;  that  is,  the  sum  of  the  positive 
terms  becomes  equal  to  the  sum  of  the  negative  terms.  But  if 
— a  be  substituted  for  x  in  equations  (2)  and  (3),  the  numerical  val- 
ues of  the  terms  of  these  equations  will  be  equal  to  the  values  of 
the  corresponding  terms  of  equation  (1),  while  tho  signs  of  the 
terms  in  equation  (2),  if  m  is  an  even  number,  will  be  the  same,  and 
those  of  equation  (3),  opposite  to  the  signs  of  the  terms  of  like 
degree  in  equation  (1).  If  m  is  an  odd  number,  the  reverse  will  be 
true  in  respect  to  signs.  In  cither  case  however,  if  a  is  a  root  of 
equation  (1),  — a  is  a  root  of  both  equation  (2)  and  equation  (3). 

An  obvious  consequence  of  this  proposition  is,  that  the  roots  of 
an  equation  are  not  affected  by  changing  the  signs  of  all  its  terms. 

EXAMPLES. 

1.  The  roots  of  the  equation  x9 — 7sc2+13os— 3=  0,  are  3,  2+ j/3. 
and  2—1/3;    what  will    be  llie  roots   of   the   equation  x3 

2.  The    roots  of   the  equation  x* — 3 

84 


398  PROPERTIES    OF    EQUATIONS. 


1,   —  2,    2-{-l/^5,  and  2—  V~b  j    what  are  the  roots  of    the 
equation  x4+3xs-f  3x8—  17  x—  18  =  0  ? 


—  — 


coefficients  of  an  equation  be  real  and  rational, 
surd  and  imaginary  roots  can  enter  the  equation  only  by  pairs. 

Let  the  coefficients  A,  J5, . . . .  U,  of  the  equation 

xm+  Ax™-1  -f-  Ex™---}- Tx-{-  U=  0,  (1) 

be  all  real  and  rational,  and  suppose  that  a  it  l/+b  is  one  of  the 
roots  of  this  equation. 

Substituting  this  value  for  x,  we  have 

=  0.  (2) 

Expanding  the  several  terms  of  this  equation  by  the  binomial 
formula,  we  have 


q 

2)a— •  •  ^±6+^(771—2)  -— 

2! 


observing,  in  reference  to  the  final  terms,  ±(K  ±:2>)mj  dbC^dz 
etc.,  that  the  sign  it  is  to  be  used  before  those  only  which  have 
odd  numbers  for  their  exponents ;  when  the  exponent  is  even,  the 
plus  sign  is  to  be  understood. 

If  the  root  of  equation  (1)  be  «-f-|/6,  the   aggregate  of  these 
developments  will  be  composed  of  two  parts,  the  one  rational  and 
the  otherurd.     The  rational  part  will  be  the  algebraic  sum  of  those 
the  even  powers  of  -\/b  for  factors,  the  zero  pow- 
er  bei^l         •*  Represent  this  part  by  M. 


le  other^urd.  Th 
irnfl  BMt^"0  ^ 
rbe^H  ifeS?» 


PROPERTIES    OF    EQUATIONS.  399 

The  irrational  part  will  be  the  algebraic  sum  of  the  terms  having 
the  odd  powers  of  y'b  for  factors.  But  since  (j/&)3  =  b^/b, 
(j/&)6  =  ^>2|/&,  etc.,  the  different  terms  of  this  part  can  be  repre- 
sented by  a  single  term  of  the  form  N^/b^  N  being  the  algebraic 
sum  of  the  coefficients  of  y/7>.  Hence  equation  (2),  under  the  sup- 
position that  a-f- j/6  is  a  root  of  equation  (1),  becomes 


which  can  be  true  only  when  we  have  separately  M  =  0,  N=  0; 

In  reducing  equation  (2)  to  equation  (3),  the  upper  signs  in  the 
expansions  of  the  terms  of  equation  (2)  were  used.  If  the  lower 
signs  in  the  equation  and  the  expansions  of  its  terms  be  used, — which 
is  equivalent  to  supposing  a — i/b  to  be  a  root  of  equation  (1), — tho 
reduced  equation  will  be 

M—N^/b  =  0.  (4) 

in  which  M  and  N  are  evidently  the  same  as  in  equation  (3).  Hence 
if  equation  (1)  has  a  root,  a-\-]/b,  ^  ^as  a^so  ^e  r00^  a — V  &• 

Now  let  us  suppose  that  a-\-V  — i  is  a  root  of  eq.  (1);  then  since 
the  even  powers  of  V — b  are  real  and  the  odd  powers  imaginary, 
the  developed  first  member  of  eq.  (2)  will  be  composed  of  two  parts, 
the  one  real  and  the  other  imaginary.  Represent  the  real  part  by  M'. 

The  imaginary  part  is  the  algebraic  sum  of  the  terms  having  the 
odd  powers  of  1/—b  for  factors.  But  since  (I/ — 6)3  =  V  b*  (  — b) 
=  bV— b,  (V7^))6  =  V  bT(^b)  =  b*V~b,  etc.,  the  different 
terms  of  this  part  can  be  reduced  to  a  single  term  of  the  form 
N'v — b.  Hence,  under  the  supposition  that  a-\-V — b  is  a  root 
of  equation  (1),  equation  (2)  becomes 

M'-\-N'V^b  =  0,  (5) 

which  requires  that  we  have  separately  M'  =  0,  N'  =  0 ;  (367). 
By  using  the  lower  signs  of  the  terms  and  their  expansions  in  equa- 
tion (2), — which  supposes  a — V — b  to  be  a  root  of  eq.  (1), — we  find 

and  by  a  simple  inspection  of  the  expanded  terms  of  equation  (2), 
we  see  that  M'  and  N'  in  equations  (5)  and  (6)  are  the  same. 

Whence  we  conclude  that  if  equation  (1)  have  a  root,  a  J-  K — b, 
it  has  also  the  root,  a — 1  — b. 


400  PKOPERTIES    OF    EQUATIONS. 


RULE  OF  DES  CARTES. 

446.  An  equation  can  not  have  a  greater  number  of  positive 
roofs  than  there  are  variations  in  the  signs  of  its  terms,  nor  a  greater 
number  of  negative  roots  than  there  are  permanences  of  signs. 

NOTE.—  A  variation  is  a  change  of  sign  in  passing  from  one  term  to 
another  ;  a  permanence  occurs  when  two  successive  terms  have  the  same 
sign.  It  is  obvious  that  the  number  of  variations  and  permanences  taken 
together  must  be  equal  to  the  number  of  terms,  less  1. 

Let  the  signs  of  the  terms  of  an  equation  be 

+     +  +  +     +       -> 

the  second,  sixth,  and  eighth  terms  giving  permanences,  and  the 
other  terms  variations. 

To  introduce  a  new  positive  root  into  the  equation,  we  must  mul- 
tiply the  equation  by  some  binomial  factor  in  the  form  of  x  —  a  ; 
and  the  signs  of  the  partial  and  final  products  will  be  as  follows  : 

1st.       2d.       3d.      4th.      5th.      6th.      7th.      8th.      9th. 


Now  we  observe,  in  the  final  result,  that  the  2d,  6th,  and  8th 
terms,  or  the  terms  which  give  permanences  in  the  proposed  equa- 
tion, are  ambiguous  ;  consequently,  let  these  ambiguous  signs  be  ta- 
ken as  they  may,  the  number  of  permanences  has  not  been  increas- 
ed. But  the  number  of  terms  has  been  increased  by  1  ;  hence,  the 
number  of  variations  has  been  increased  by  1,  at  least. 

Again,  to  introduce  a  new  negative  root  into  the  proposed  equa- 
tion, we  must  multiply  by  some  binomial  factor  in  the  form  of  x-\-a' 
and  the  partial  and  final  products  will  be  as  follows  : 

1st.        2d.       3d.       4th.      5th.      6th.      7th.      8th.      9th. 

+    +  +  +    4- 

_  +4--    +    --    +    +    - 

+     +     ±     ±     ±  ±     +     ± 

Her.c,  in  the  final  result,  the  3d,  4th,  5th,  7th  and  9th  terms,  01 
the  terms  which  give  variations  in  the  proposed  equation,  are  am- 

V 


CARDAN'S  RULE.  401 

biguous ;  consequently,  the  number  of  variations  has  not  been  in- 
creased. But  the  number  of  terms  has  been  increased  by  1 ;  hence, 
the  number  of  permanences  has  been  increased  by  1,  at  least. 

Thus  we  have  shown  that  the  introduction  of  each  positive  root 
must  give  at  least  one  additional  variation,  arid  the  introduction  of 
each  negative  root  must  give  at  least  one  additional  permanence. 
Hence  the  whole  number  of  positive  roots  can  not  exceed  the  num- 
ber of  variations,  and  the  whole  number  of  negative  roots  can  not 
exceed  the  number  of  permanences;  the  proposition  is  therefore 
proved. 

44  7.  Although  the  introduction  of  a  positive  root  -will  always 
give  an  ;  dditional  variation  of  signs,  it  is  not  true  that  a.  variation 
9f  signs  in  the  terms  of  an  equation  necessarily  implies  the  presence 
of  a  real  positive  root.     Thus,  the  equation, 
x3— x9— T.r-f-15  =  0 

has  2  variations  of  signs,  and  1  permanence.     But  its  roots  are 
2-fi/Hl,    2— l/~l,   and    —3, 

no  one  being  positive  and  real. 

But  ichen  the  roots  are  all  real,  the  number  of  positive  roots  is 
equal  to  the  numler  of  variations,  and  the  number  of  negative  roots 
is  equal  to  the  number  of  permanences. 


CARDAN'S  RULE  FOR  CUBIC  EQUATIONS. 

.  It  has  been  shown,   (437"),  that  any  equation  can  be 
transformed  into  another  which  shall  be  deficient  of  its  second  term 
That  is,  every  cubic  equation  can  be  reduced  to  the  form  of 
.         x*+3px  =  2q>  (1) 

and  the  solution  of  this  equation  must  involve  the  general  solution 
of  cubics.     We  make  3/>  the  coefficient  of  x,  and  2q  the  absolute 
term,  in  order  to  avoid  fractions  in  the  following  investigations: 
Assume  x  =  v-\-y  ;  then  cq.  (1)  becomes 


Expanding  and  reducing,  we  have 

*3+ 
34* 


402  PROPERTIES    OF   EQUATIONS. 

Now  as  the  division  of  x  into  two  parts  is  entirely  arbitrary,  we  are 
permitted  to  assume  that 

vy+p  =  o  >  (4) 

whence,  from  eq.  (3),  v*-{-y*  =  2q.  (5) 

If  we  obtain  the  value  of  y  from  (4),  and  substitute  it  in  (5),  we  shall 
have,  after  reducing, 

whence,  v*  =  q±3/<f-\-p*.  (7) 

Substituting  this  value  of  v*  in  (5),  we  have 


But  by  hypothesis  x  =  v-\-y  ;  hence,  taking  the  sum  of  the  cube 
roots  of  (i)  and  (8),  we  have 


which  is  Cardan's  formula  for  cubic  equations. 

44O.  When  p  is  negative,  in  the  given  equation,  and  its  cube 
numerically  greater  than  (f,  the  expression  V  tf+p*  becomes  imag- 
inary ;  this  is  called  the  Irreducible  Case.  We  must  not  conclude, 
however,  that  in  this  case  the  roots  of  the  equation  are  imaginary  ; 
for,  admitting  the  expression  V  <f-\-p*  to  be  imaginary,  it  can  be 
represented  by  aV  —  1  ;  whence  the  value  of  x  in  formula  (A)  be- 
comes 


x  = 


W 
I 


or,  *   =    l+  V-i*+    l-  V-!l         © 


Now  by  actually  expanding  the  two  parts  in  tlie  second  member 
of  (3),  and  adding  the  results,  the  terms  containing  V  —  1  will  can 
eel,  and  the  final  result  will  be  real.  Hence,  in  the  irreducible  case 
all  the  roots  of  the  equation  are  real  ;  formula  (A)  is  therefore  prac- 
tically applicable  only  when  two  of  the  roots  are  imaginary.  In  this 
case  the  real  root  can  be  found  directly  by  the  formula  ;  the  equa- 
tion may  then  be  depressed,  by  division,  to  a  quadratic,  which  wiU 
give  the  two  imaginary  roots. 


CARDAN'S  RULE.  403 


EXAMPLES. 

1.  Find  the  roots  of  the  equation,  f  ^t  T"-  '" 

20  =  0. 


To  transform  this  equation  into  another  deficient  of  its  2d  term, 
according  to  (437),  put  aj=y-f~'Ij  and  we  shall  have  for  the 
transformed  equation, 

*•-!*  =  W- 

To  apply  the  formula  to  this  equation,  we  have 
3p  =  —  I,  OTp  =  —l', 
22  =  V74,  or  2  =  *tf. 


=  ic'  =  ±  W- 

=  (Vtfifc  W^+CW  +  W/ 


x  =  |+|  =  5,  the  real  root. 
Dividing  the  given  equation  by  x  —  5,  we  obtain  for  the  depressed 

equation  « 

x»—  2x-}-4  =  0; 

whence,  x  =  \±V  —  3. 

Hence  the  three  roots  arc  5,  \-\-V  —  3,  and  1  —  V  —  3. 

2.  Given  x*+Qx  =  88,  to  find  the  values  of  x. 
To  apply  the  formula,  we  have 

3p  =    G,         or  p  =    2  ; 
2q  =  88,        or  q  =  44. 

whence,  Vf+p*  =  1/1944+8  =  ±44.090815+. 

And  we  have 

x  =  (44+44.090815)*-K44—  44.090815)*; 
or,  x  =  4.4495  —  .4495  =  4,  the  real  root. 

The  depressed  equation  will  be  xa-J-4x  —  22  =  0;  whence 

x  =  —  2±31/H2; 
and  the  three  roots  are     4,  —  2-J-81/H2,  and  —  2—81/^2. 

3.  Given  x*  —  Qx  =  5.6,  to  find  one  value  of  x. 

This  example  presents  the  irreducible  case  ;  the  solution,  by  the 
method  of  series,  is  as  follows  : 


4:04  PROPERTIES    OF    EQUATIONS. 

TVc  have  p  =  —2,     q  =  2.8  ;  hence, 


x  =  (2.8+1/7^4=8)+  (2.8—  1/L 
or,  2  =  (2.8+.4l/^I*+2.8--.41/= 


Put          6  =  4  i/Hi  .  then  6*  =  —  ^,  i«  =  ^  X  ,V 

Also,       (I-I-IT/-I)*  =  (1+6)*  ;     (l-4T/=!  l)*=(l-i)* 
By  the  binomial  theorem 

i  122-5  2-5-8 

(1+6)*  =  1+36-  ^+  ^-^'_  3^^-+  .... 

i  122-5  2-5-8 

(l_t)*  =  1-  ?i_  g-gy,  —  6y_  333^'-  .... 


Sum  =2 

=  2+.004535—  .000034  =  2.004569. 

Hence,  we  have 

« 

-a7=='  2.004569;  ^  =  (2.004569)1^^8  =  2.82535,  Am. 

4.  Given  re*  —  Qx  —  6  =  0,  to  find  one  value  of  x. 

Ans.  x  =  f  2+^4  =  2.8473+. 

5.  Given  x*-\-9x  —  6  =  0,  to  find  one  value  of  x. 

Ans.  x  =  ty  9+lf  ~3  =  .63783+. 

6.  Given  a8+6:ca—  13S+24  =  0,  to  find  the  values  of  x. 

Ans.  x  =  —8,  1+1/H2,  or  1—1/-32, 


NUMERICAL   EQUATIONS    OF 


SECTION  IX. 

SOLUTION  OF  NUMERICAL  EQUATIONS  OF  HIGHER 
DEGREES. 

LIMITS  OF  REAL  ROOTS. 

4«5O.  All  positive  roots  of  an  equation  are  comprised  between  0 
and  -f-  oo,  and  all  negative  roots  between  0  and  —  GO.  But  in  tho 
solution  of  numerical  equations  of  higher  degrees,  it  is  necessary 
to  be  able  at  once  to  assign  much  narrower  limits.  As  preliminary 
to  this,  we  will  first  show  how  an  equation  is  affected  by  substituting 
for  the  unknown  quantity  numbers  greater  or  less  than  the  roots, 
and  numbers  between  which  the  roots  are  comprised. 

4«5I.  If  an  equation,  in   its  general   form,  be  regarded   as  the 
product  of  the  binomial  factors  formed  by  annexing  the  roots,  with 
their  opposite  signs,  to  .r,  we  observe  that  the  siyii  of  this  product 
can   not  be   affected  by  the  imaginary  roots.      For,  according  to 
(445),  if  an  equation  have  one  root  in  the  form  of  a-j-V  —  6,  it 
will  have  another  in  the  form  of  a  —  V  —  b.     But  we  have 
(x—a-  V—  ~b)  (x—  a+V~b)  =  (x—  «)3-f  6, 
a  result  which  is  in  all  cases  positive. 

453.  Let  a,  6,  c,  d,  etc.,  be  the  real  roots  of  an  equation,  ar- 
ranged in  the  order  of  their  algebraic  values;  then  the  equation  may 
be  represented  as  follows  : 

(a-—  a)(ar—  i)(a;—  c)(o-—  rf)  ____  =0. 

If  we  substitute  h  for  x,  the  first  member  will  become 


Now  if  h  be  less  than  the  least  root,  a,  every  factor  will  be  neg- 
ative ;  and  the  whole  product  will  be  positive  or  negative,  according 
as  the  number  of  factors  is  even  or  odd.  But  the  number  of  factors 
is  equal  to  the  degree  of  the  equation,  (4^7);  hence, 

1.  —  If  a  number  less  than  the  least  root  be  substituted  for  x  in  an 

A* 


406         NUMERICAL    EQUATIONS    OF   HIGHER   DEGREES. 

equation,  the  result  will  be  positive  when  the  equation  is  of  an  even 
degree,  and  negative  when  the  equation  is+of  an  odd  degree. 

Again,  if  h  be  greater  than  the  greatest  root,  then  every  factor  will 
be  positive,  and  consequently  the  whole  product  positive.  Hence, 

2. — If  (t  number  greater  than  the  greatest  root  be  substituted  for 
x  in  an  equation,  the  result  will  in  all  cases  be  positive,. 

Still  again  j  suppose  h  to  be  at  first  less  than  a,  but  afterward 
greater  than  a  and  less  than  b.  This  change  in  the  value  of  h  will 
change  the  sign  of  the  factor  (h — a),  and  consequently  the  sign  of 
the  whole  product.  If  in  the  next  place  h  be  made  greater  than  b 
but  less  than  c,  the  sign  of  the  product  will  be  changed  again,  for 
the  same  reason  as  before.  And  in  general,  there  must  be  a  change 
of  sign  every  time  the  variable  h  passes  the  value  of  a  real  root  of 
the  equation,  and  at  no  other  time.  Hence, 

3. — If  when  two  numbers  are  substituted  in  succession  for  x,  in 
an  equation,  the  results  have  contrary  signs,  there  must  be  at  least  one 
real  root  included  between  these  numbers. 

It  may  be  observed,  also,  that  if  one,  three,  Jive,  or  any  odd  num- 
ber of  roots  be  included  between  the  two  numbers  substituted,  the 
results  will  show  a  change  of  signs.  But  if  an  even  number  of  roots 
be  included,  there  will  be  no  change  of  signs. 

4:*>3.  If  P  denote  the  numerical  value  of  the  greatest  negative 
coefficient  in  an  equation,  and  n  the  number  of  terms  which  precede 
the  first  negative  coefficient,  then  VP-|-1  will  be  a  superior  limit  of 
the  positive  roots  of  this  equation. 

Let      ./m-h  A^-'-f  Bxm~*-\-  Or"1-8-}- -f  Tx+  U—  0.      (1) 

If  we  omit  those  positive  terms,  if  any,  which  occur  between  xm  and 
the  first  negative  term,  and  then  put  — P  for  the  coefficient  of  every 
other  term  after  xm,  we  shall  have  f*3\ 

,t  »_(p.<«+JRc— — »+  ....  ^Px-fty^  0.  (2) 

Now  it  is  evident  that  any  value  which,  substituted  for  x,  will 
give  a  positive  result  in  eq.  (2)  will  give  a  positive  result  also  in  eq. 
(1).  For,  the  sum  of  all  the  negative  terms  in  (1)  can  not  possibly 
be  greater  than  the  negative  part  of  (2) ;  besides,  there  may  be  one 
or  more  positive  terms  in  (1)  which  are  omitted  in  (2). 


LIMITS    OF    REAL    ROOTS. 
Dividing  every  term  of  (2)  by  xm,  we  obtain 


Make  x  —  \/P-}-l  —  r-fl,  where  r  is  put  in  the  place  of  '(/P 
for  the  sake  of  simplicity.     Remembering  that  P  =  rn,  we  have 


Summing  the  geometrical  series   found  in   the  parenthesis,  by 
^  (-??')]  the  equation  becomes 

r     \n~l 

H)  (4) 

Now  since  ,  is  less  than  unity,  the  expression  which  consti- 
tutes the  first  member  of  (4)  is  positive.  Moreover,  the  negative 

(r    \n~l 
—  1     ,  must  always  be  less  than  unity,  whatever  be  the 

value  of  r  ;  hence,  no  value  of  r,  however  great,  will  render  the  first 
member  of  (4)  negative.  Thus  we  have  shown  that  if  we  substitute 
for  x  the  quantity  VP  +  1,  or  any  greater  value,  the  result  will  be 
positive  in  equation  (3)  ;  the  result  will  therefore  be  positive  in  cq. 
(•2),  and  also  in  eq.  (1).  Hence,  by  (4r*IS?  2\  \AP-r-l  is  a  superior 
limit  of  the  positive  roots  in  any  equation,  which  wr.s  to  be  proved. 

In  applying  the  principle  just  established,  the  absolute  term  must 
be  regarded  as  the  coefficient  of  x9  ;  and  if  the  equation  is  incom- 
plete, the  deficient  terms  must  be  counted,  in  finding  n. 

It  should  be  observed  also,  that  an  equation  having  no  negative 
term  can  have  no  positive  roots.  For,  every  positive  number  sub- 
sti+uted  for  x  will  render  the  first  member  positive.  That  is,  no 
positive  value  of  x  can  reduce  the  first  member  to  zero. 


EXAMPLES. 


1.  Find  the  superior  limit  of  the  positive  roots  of  the  equation 

_f  fu;«_}_2x8_  14x3— 2Cxr-f  10  =  0. 

llcre  n  =  3  and  P  =  26.     Hence  we  have,  in  whole  numbers, 

=  4,   An*. 


408         NUMERICAL    EQUATIONS  OF    HIGHER    DEGREES. 

2.  Find  the  superior  limit  of  the  positive  roots  of  the  equation 
x45x3—  25xa—  12a:-68==0.  Ans.  6. 


3.  Find  the  superior  limit  of  the  positive  roots  of  the  equation 
a:4—  5.*'—  93+12  =  0.  Ans.  4. 

4.  Find  the  superior  limit  of  the  positive  roots  of  the  equation 
x*+x*+3x—8  —  0.  Ans.  3. 

454.  To  determine  the  superior  limit  of  the  negative  roots  of 
an  equation,  numerically  considered, 

Change  the  signs  of  the  alternate  terms,  c  mnting  the  deficient 
terms  when  the  equation  is  incomplete;  then,  apply  the,  preceding  rule. 

For,  according  to  (444),  the  positive  roots  in  the  new  equation 
will  be  numerically  the  negative  roots  in  the  given  equation. 

EXAMPLES. 

1.  Find  the  superior  limit  of  the  negative  roots  of  the  equation 
a'—  3.c«+5:e+7  =  0.         Ans.  ^7-f  1  =  3,  in  whole  numbers. 

2.  Find  the  superior  limit  of  the  negative  roots  of  the  equation 
rr4—  15.*;2—  lO.r+24  =  0.  Am.  5. 

3.  Find  the  superior  limit  of  the  negative  roots  of  the  equation 
X«__3x*_j-2.r4+27xs—  4xa—  1  =  0.  Ans.  4. 

LIMITING  EQUATION. 

4«5t5.  If  there  be  one  equation  whose  roots,  taken  in  the  order 
of  their  values,  are  intermediate  between  the  roots  of  another,  the 
former  is  said  to  be  the  limiting  equation  of  the  latter. 

/JoG.  Any  equation  being  given,  its  limiting  equation  may  be 
formed  Ly  putting  its  first  derived  polynomial  equal  to  zero. 

If  a,  6,  c,.  .  .  .&,  I  are  the  roots  of  the  given  equation  X  =  0,  and 
a',  &',  c',  .  .  .  .  kr  are  the  roots  of  the  derived  polynomial  X^  =  0, 
each  set  being  arranged  in  the  order  of  their  values,  then  we  are  to 
show  that  all  these  roots,  taken  together,  and  arranged  in  the  order 
of  their  values,  will  be  as  follows  : 


In  both  equations,  put  x  =  z'+w,  developing  the  terms,  and  ar- 


LIMITING    EQUATION.  409 

ranging  the  results  according  to  the  ascending  powers  of  u.  Ob- 
serve that  JTo  is  the  first  derived  polynomial  of  Xl ;  hence,  adopt- 
ing the  same  notation  as  in  (43O),  we  have,  from  the  two  equations, 

x  =  x*  -MV-f  -^'-f-      *3+  .  • . .  =  o,        (i) 


Xl  =  X'  1 4- A'2«+  -:_3wa+  ±-*i*'+ . . . .  =  0 ;         (2) 

where,  it  will  be  observed,  A7,  A'13  AT'2,  etc.,  represent  what  X,  ATj, 
Ao,  etc.,  become,  when  x'  takes  the  place  of  x. 

Now  suppose  x'  =  r ;  that  is,  a;  i=  r-|-w,  r  being  any  rootf  o/  the 
git- e n  equation.  Then  X'  =  0 ;  and  as  Ar'u  A7^,  A/3,  now  receive 
definite  values,  the  values  of  X  and  A\  may,  or  may  not  become 
zero  by  giving  a  particular  value  to  u.  Dropping  X'  from  (1),  and 
factoring  the  result,  we  have 

X   =  u(  X'.+  ^u+^u*4-  ....},  (3) 

(4) 

where  the  different  terms  may  be  essentially  positive  or  negative, 
according  to  the  values  or  r  and  w,  upon  which  they  depend. 

Now,  it  is  evident  that  by  causing  u  to  diminish  numerically,  each 
term  after  the  first,  in  the  parenthesis,  may  be  made  as  small  as  we 
please  ;  and  by  making  u  sufficiently  small,  the  sum  of  the  terms 
containing  u,  in  each  parenthesis,  may  be  made  less,  than  the  first 
term  X' , ;  in  which  case  the  essential  sign  of  the  quantity  in  either 
parenthesis  will  depend  upon  the  sign  of  X'  j .  Thus,  when  u  is 
indefinitely  small,  the  signs  of  the  functions,  A" and  A,,  will  depend 
upon  the  signs  of  u  (A7  j)  and  X7 n  respectively.  Hence,  when  u 
is  negative,  A^and  A\  will  have  opposite  signs;  but  when  u  is  pos- 
itive, X  and  A'1  will  have  the  same  signs. 

/1«57'.  Thus  we  have  shown,  that  if  we  substitute  in  a  given  equa- 
tion X  =  0,  and  its  first  derived  polynomial  Xj  =  0,  a  quantity 
r — u.  which  is  insensibly  less  than  the  root  r,  the  results  will  have  op- 
posite signs  ;  but  if  we  substitute  the  quantity  r-f-u,  which  is  insensibly 
greater  than  the  root  r,  the  results  will  have  the  same  sign. 

408.  Consider  the  quantity  substituted  in  the  two  functions  to 
be  insensibly  less  than  a,  the  least  root  of  X  =  0,  and  let  it  increase 
35 


410         NUMERICAL    EQUATIONS   OF    HIGHER    DEGREES. 

till  it  is  insensibly  greater  than  a.  In  passing  the  root  a.  the  func- 
tion X  will  change  sign,  (4529  3)  ;  hence  the  signs  of  the  func- 
tions will  be  as  follows  : 

X    X,  X    Xl 

rx  —  a—u     -f      —  ,  —      -f, 

For  }'x  $sa  0      —  ,    or  else    0      -J-, 

\x=a+u    —    —  ,  +     +. 

Now  let  the  substituted  quantity  increase  from  x  =  a-\-u  to 
x  =  b  —  «,  a  value  insensibly  near  to  b,  the  next  root  of  X  =.  0. 
According  to  the  principle  already  established  (4-57),  X  and  Xl 
must  now  have  opposite  signs.  And  since  X  can  not  have  changed 
its  sign  during  the  change  of  x  from  <t-\-u  to  b  —  u,  there  must  have 
been  a  change  of  sign  in  the  function  X}.  Hence,  by  (452,3) 
one  root  of  X}  --=0  is  found  between  a-\-u  and  b  —  u.  or  between  a 
and  b.  In  like  manner,  it  can  be  shown  that  Xj  =  0  has  one  root 
between  b  and  c,  one  between  c  and  d,  and  so  on.  Ilcncc  the  prop- 
osition is  proved. 


STURM'S  THEOREM. 

.  The  object  of  Sturm's  Theorem  is  to  determine  the  num- 
ber of  the  real  roots  of  an  equation,  and  likewise  the  places  of  these 
roots,  or  their  initial  figures  when  the  roots  are  irrational. 

NOTE.  —  This  difficult  problem,  which  for  a  long  time  baffled  the  skill 
of  mathematicians,  was  first  solved  by  M.  Sturm,  his  solution  being  sub- 
mitted to  the  French  Academy  in  1829. 

4@4>.  We  have  seen,  (435),  that-the  equal  roots  of  an  equa- 
tion may  always  be  found  and  suppressed.  Now  let 

X  =  xr+Ax^+Bx™-*-^  ____  Tr-\-u  =  0 

represent  any  equation  having  no  equal  roots,  and  X1  —  0  its  first 
derived  polynomial,  or  its  limiting  equation. 

V\7e  will  now  apply  to  the  functions,  JCand.JTn  a  process  similar 
to  that  required  for  finding  their  greatest  common  divisor  (1O5), 
but  with  this  modification,  namely;  that  ice  change  the  signs  of  the 
siii'i-wicr  remainders,  and  neither  introduce  nor  reject  a  negatibe 
factor,  in  pn  paring  for  division. 

Denote  the  successive  remainders,  with  their  signs  changed,  by 


STURM'S  THEOREM.  411 

\ 

R,  RIJ  R.2,. . .  .J^n-n  ^n-  Since  the  given  equation  has  no  equal 
roots,  there  can  be  no  common  divisor  between  X  and  Xl,  (435); 
hence,  if  the  process  of  division  be  continued  sufficiently  far,  tho 
last  remainder,  Rn,  must  be  different  from  zero,  and  independent  o/x. 

Now  in  the  several  functions,  .£,  Xl>  R,  M\^  R*,-  •  •  •Rn—\y  RM 
let  us  substitute  for  x  any  number,  as  A,  and  having  arranged  the 
sig;i:i  of  the  results  in  a  row,  note  the  number  of  variations  of  signs. 
Next  substitute  for  x  a  number,  7t',  greater  than  7i,  and  again  note 
the  number  of  variations  of  signs.  The  difference  in  the  number  of 
variations  of  signs,  resulliny  from  the  two  substitutions,  will -be  equal 
to  the  number  of  real  roots  comprised  between  h  and  h'. 

This  is  Sturm's  Theorem,  which  we  will  now  demonstrate. 

Let  Q,  Qi,  (?;>> On-i»  Qn  denote  the  quotients  in  the  succes- 
sive divisions.  Now  in  every  case,  the  dividend  will  be  equal  to  tho 
product  of  the  divisor  and  quotient,  plus  the  true  remainder,  or 
the  remainder  with  its  sign  changed.  Hence, 

(1)  X  =  Xl  Q  —  R 

(2)  X}  =  R  Q,  - 

(3)  R  =^Ca  -      av 

(4)  R  =  R.Q  ' 


From  these  equations,  it  follows, 

1 — If  any  number  be  substituted  for  x  in  the  functions  X,  Xn 
R7  RJ,.  . .  .Rn,  no  two  oj  them  can  become  zero  at  the  same  time. 

For,  if  possible,  let  such  a  value  of  h  be  substituted  for  x  as  will 
render  Xl  and  R  zero  at  the  same  time.  Then  the  second  equation 
of  (A)  will  give  R ,  =  0 ;  whence,  the  third  equation  will  become 
R.2  =  0 ;  and  tracing  the  series  through,  we  shall  have,  finally, 
Rn  =  0,  which  is  impossible. 

2. — If  any  one  of  the  functions  become  zero  by  substituting  a 
particular  value  for  x,  the  adjacent  functions  will  have  contrary 
signs  far  the  same  value. 

For,  suppose  R  l  in  the  third  equation  to  become  zero  ;  then  this 
equation  will  reduce  to  R  =  — R.2.  That  is,  R  and  /?._,  have  con- 
trary signs. 

Having  established  these  principles,  suppose  the  quantity  h,  which 


412         NUMERICAL    EQUATIONS    OF    HIGHER    DEGREES. 

is  to  be  substituted  simultaneously  in  all  the  functions,  to  be  a  vari- 
able, changing  by  insensible  degrees  from  a  less  to  a  greater  value. 
As  it  passes  any  of  the  roots,  the  function  to  which  this  root  be- 
longs will  reduce  to  zero,  and  change  sign,  (45S,  3). 

Let  p  be  a  little  less  than  a  certain  root  of  7?.,,  and  q  a  little 
grea'tcr  than  the  same  root,  the  two  values  being  so  taken,  however, 
that  no  root  of  R,  or  R3  shall  be  comprised  between  them,.  As  h 
changes  fromjp  to  q,  R2  will  reduce  to  zero,  and  change  sign.  But 
neither  JKl  nor  E3  will  change  sign;  and  since,  according  to  the 
second  principle,  these  functions  have  opposite  signs  when  R%  =  0, 
they  must  have  opposite  signs  also  when  h  =  p  or  h  —  q.  Now 
when  h  =p,  the  arrangement  of  signs  must  be 

Rl     R.2     R3  R^     R2    R3 

+      ±      -      or     ,-      ±      +; 

giving  one  variation  and  one  permanence,  whichever  way  the  double 
sign  be  taken.     When  h  —  q  the  signs  must  become 
Rl     7?2     R3  R1     R.2     R3 

+       T      — ,     or  =F      +; 

giving,  as  before,  one  variation  and  one  permanence,  so  that  tho 
whole  number  of  variations  is  neither  increased  nor  diminished. 

This  reasoning  obviously  applies  to  any  function  which  is  situated 
between  tico  other  functions.  Hence, 

3. —  When  h  passes  a  root  of  any  function  intermediate  between 
X  and  Rn ,  the  number  of  variation^  of  signs  will  not  be  altered. 

As  the  last  function,  Rm  is  independent  of  x,  its  sign  will  not  be 
changed  by  any  substitution  for  x.  It  follows,  therefore,  that  if  any 
change  is  produced,  in  the  number  of  variations  of  sign's,  it  must  re- 
sult from  the  alternation  of  signs  in  the  original  function  X. 

Let  a,b,c,d,  ....  I  be  the  roots  of  JT,  taken  in  the  order  of  their 
values.  Then  the  roots  of  Xl  will  be  found,  the  first  between  a 
and  6,  the  second  between  b  and  r,  and  so  on  ;  (458).  The  de- 
gree of  Xl  is  less  by  1  than  the  degree  of  X;  hence,  if  the  degree 
of  X  is  odd  the  degree  of  Xl  will  be  even,  arid  if  the  degree  of  X 
is  even  the  degree  of  X^  will  be  odd.  Now  take  h  less  than  a  ;  ac- 
cording to  (4:512,  1),  the  signs  of  X  and  Xl  will  be  unlike,  giving 
a  variation.  Let  h  increase  till  it  is  insensibly  greater  than  a  ;  X 
will  change  sign,  and  the  variation  between  X  and  Xl  will  be  lost. 


STURM'S  THEOREM.  413 

Now  let  h  increase  till  it  is  insensibly  loss  than  It.  Tt  will  pnss  the 
first  root  of  Xi,  causing  the  signs  of  X  and  Xl  to  be  again  unlike; 
but  by  (3;,  this  change  in  the  sign  of  Xl  will  not  alter  the  whole 
number  of  variations  in  the  signs  of  the  functions.  Again  let  h 
increase  till  it  is  insensibly  greater  than  b  ;  X  will  again  change  sign, 
and  another  variation  will  be  lost.  In  like  manner  it  may  be  shown 
that  the  number  of  variations  will  be  diminished  by  1  evert/  time  h 
p  ses  a  root  »f  X;  hence  the  truth,  of  the  theorem. 

4O1.  If  we  substitute  for  x  in  the  several  functions  h  .  •=.  —  QQ 
and  h'*=  -\-  oo,  successively,  we  shall  determine  at  once  the  whole 
number  of  real  roots  in  the  given  equation.  To  ascertain  the  signs 
of  the  functions  resulting  from  these  substitutions,  we  require  the 
following  principle  : 

If  in  am/  polynomial  involving  the  descending  powers  of  x,  in- 
finity be  substituted  for  x,  the  sign  of  the  whole  exf>ression  will  de- 
pend upon  the  sign  of  the  first  term. 

Let  Axm-\-Bxm-l-\-Cxm-^Dxm-y  -j-  j^e"-4-}-  .  .  .  .  bo  the  given 
polynomial,  if  x  =  GO,  then 


because  every  term  in  the  second  member  is  less  than  any  assigna- 
ble quantity,  or  zero,  (188,  2).  Multiplying  both  members  of  (1) 
by  xmj  we  have 

Axm  >  Bxn-l+  Cx^+Dx^+Ex™-*-}-  ...  (2) 
That  is,  when  x  —  GO,  the  first  term  of  the  given  polynomial  is 
num;rio:iHy  greater  than  the  sum  of  all  the  other  terms.  Hence 
the  sign  of  the  whole  will  be  the  same  as%  the  sign  of  the  first  term. 
4LG2.  In  the  application  of  Sturm's  Theorem,  we  may  always 
suppress  any  numerical  factor  in  any  of  the  functions  Xlt  R,  Itlt 
etc.  j  for  this  will  not  affect  the  sign  of  the  result. 

1.  Given  the  equation  x*—  3x2  —  12x-|-24  =  0,  to  find  the  nurn 
bcr  and  situation  of  the  real  roots. 

Suppressing  monomial  factors,  we  have  for  tho  several  functions* 

X  =  x3  —  3xa  —  I2x  -f-  24, 

XI  =  xa  —  2x    —\ 
R    =  x    —  2, 

R,  =4. 
35* 


4J4         NUMERICAL    EQUATIONS   Oi1    HIGH  Ell    DEGREES. 

Substituting  in  these  functions  x  == — co  and  ;r  =-j-ce  succes- 
sively, we  obtain  the  following  results,  in  respect  to  signs : 

X    Xl     R    R, 

For    $x  =  —&>>  —     +     —     +>     3  variations. 


_ 

Hence,  the  given  equation  has  3     real  roots. 

Since  the  signs  in  the  given  equation  present  two  variations  and 
one  permanence,  two  of  the  roots  must  be  positive,  and  the  other 
negative,  (446).  To  ascertain  the  situation  of  the  positive  roots, 
let  us  substitute  in  the  functions,  x  =  0,  x  —  1,  x  =  2,  etc.,  suc- 
cessively, noting  the  variations  of  signs  in  t]ie  results. 

x  =  0,  signs,  -f-     —     —     -J-,     2  variations. 

x  =  l,      «       -f     —     —     -f,     2 

x  =  2j      "       —     —     ±     4-      1  variation. 

*  =  8,     « +     +,     1         « 

a;  =  4,      «       —     -f     +     +,     1 

a;  =  5,      «        -f     -j-     +     +,     0         " 
Since  one  variation  is  lost  in  passing  from  x  =  1  to  x  =  2,  and  ono 
also  in  passing  from  x  =  4  to  x  =  5,  one  positive  root  must  be  situ- 
ated between  1  and  2,  and  the  other  between  4  and  5. 

To  ascertain  the  situation  of  the  negative  root,  substitute  x  =  0, 
x  =  — 1,  x  =  — 2,  etc.  j  the  signs  are  as  follows  : 

x  =      0,  signs,  -f-     —     —     -{-,     2  variations. 

*  =  — i,  •*     +  —  —   +,  2      " 

For  /  —  2,     «       -f-     4-     —     -f,     2         « 

-3,     «       +     +     —     +,     2         « 
-4,     «       -     -j-     -     +,    3 
Hence,  the  negative  root  is  situated  between  — 3  and  — 4.     The 
initial  figures  of  the  several  roots  will  be  1,  4,  and  — 3. 

2.  Given  the  equation,  x-4_2x8— 7.«2-flOx-j-10  =  0,  to  find  the 
number  and  situation  of  its  real  roots. 
In  this  example,  we  have 

X  =  x4— 2x3— 7xs-flO^+10,    ^  =  2x9— 3xa— 7x-f  5, 
^  =  17.r2— 23x— 45,     ^  =  152x— 305,     R.2  =  524785. 
Let  x  =  — oo;  we  have     -f     —     -f-     —     -f-,     4  variations. 
«      «        -f     +     +     -f     +,     ° 


STURM'S  THEOREM.  415 

Hence,  the  roots  are  all  real.  And  since  the  signs  of  the  given 
equation  give  2  variations  and  2  permanences,  two  of  the  roots  tire 
positive  and  two  are  negative.  To  find  the  situation  of  the  roots, 

,     2  variations. 
,     2 
,     2 
,     0 

i     3 

,     3          « 
,    4 

Hence,  there  are  two  roots  situated  between  2  and  3,  one  between  0 
and  — 1,  and  one  between  — 2  and  — 3. 

We  wish  now  to  find  limits  which  will  separate  the  two  roots  that 
lie  between  2  and  3.  Let  us  transform  the  given  equation  into 
another  whose  roots  shall  be  less  by  2.  By  (443),  the  operation 
will  be  as  follows  : 

1     —2     —7     +10     +10(2 
+2         0     -14       -  8 

0     —7    —  4,  +  2  ==  X' 
2+4—6 


Let  x  = 
x  = 
x  = 

0  ;  we  have  +     +     —     — 

1;      "        +    

9  •          «                1 
-•  J 

x  = 

3;       " 

+  +  +  + 

x  = 

-1;       « 

—  +  +  — 

X    =. 

-2;       " 

—    +    — 

X    = 

O    .              ti 

+     —     +     — 

2    —3,    —10  =  X't 
2     +8 


4-+5  =  ^2 


•    2.3 

The  transformed  equation  is  therefore 

F=y+6y'+5.y'— l(ty+2  =  0. 

Now  since  the  two  positive  roots  of  the  original  equation  are  found 
between  2  and  3,  the  two  corresponding  roots  of  the  transformed 
equation  must  lie  between  0  and  1.  The  situation  of  these  roots 
may  be  found  from  V  alone,  by  trials  as  follows : 

Substitute  x  =  0,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8. 

The  signs  of  Fare  _|_   _j_   _[_  — —  _j.. 


416         NUMERICAL    EQUATIONS   OF    HIGHER    DEGREES. 

Hence,  one  root  of  V  lies  between  .2  and  .3,  and  one  between  .7 
and  .8.  Consequently  the  initial  figures  of  the  two  positive  roots  in 
the  original  equation,  are  2.2  and  2.7. 

NOTE.  —  If  we  had  found  the  initial  figures  of  the  two  positive  roots  of 
V  to  be  the  same,  we  should  have  proceeded  to  transform  F",  and  make 
similar  trials  with  the  result. 

We  are  now  prepared  to  find  the  roots  of  an  equation  to  any  de- 
gree of  accuracy,  by 

IIORNER'S  METHOD  OF  APPROXIMATION. 

463.  In  the  year  1819,  W.  G.  Homer  Esq.,  an  English  math- 
ematician, published  a  most  elegant  and  concise  method  of  approxi- 
mating to  the  roots  of  a  numerical  equation  of  any  degree.     The 
process  consists  in  a  series  of  transformations,  the  routs  of  each  suc- 
cessive equation  being  less  than  the  roots  of  the  preceding  equation 
by  the  initial  figures  of  the   preceding  roots.     But  in  making  the 
several  transformations,  the  initial  figures   are   obtained    by   trial 
division,  as  in  square  and  cube  root,  and   not  by  substitutions,  as  in 
the  last  article. 

464.  Let  us  take  an  equation  in  the  general  form,  thus  : 
X  =  xM+Axm-l+£xm-9+  ____  -f  TJ-+  U  =  0      (1) 

Let  r  represent  the  initial  figure  or  figures  of  one  of  the  real  roots 
of  this  equation,  as  found  by  Sturm's  Theorem,  or  otherwise.  Now 
let  the  equation  be  transformed  into  another  whose  roots  shall  be 
less  by  r.  Put  x  —  r-j-y  j  we  shall  have 

V  =  y  +4y-  '4-7?y»-24-  .  .  .  .  -f-  T'y+  U  '  =  0          (2) 
In  this  equation  y  is  supposed  to  represent  a  decimal,  since  r  in- 
cludes at  least  the  entire  part  of  the  required  root.     Hence,  the 
terms  containing  the  higher  powers  of  y  are  comparatively  small  ; 
neglecting  these,  we  have,  approximately, 


Denote  the  first  figure  of  this  quotient  by  s  ;  put  y  =  s-f-z.     Trans- 
forming eq.  (2)  into  another  whose  roots  shall  be  less  by  s,  we  have 

V  =  -z™  +  A"  z™-*+B"  -.""-'*  +  .  .  .  ,  +  r'z-f  U"  =  0. 


HORNER'S  METHOD.  417 

Whence  we  have,  as  before, 


z  =  — 


where  t  is  another  figure  of  the  required  root.     This  process  may 
be  continued  at  pleasure,  and  we  shall  have,  finally, 


Hence,  to  solve  a  numerical  equation  of  any  degree,  we  first  Jind 
by  Sturm's  Theorem,  or  otkericisfo  the  number-  of  real  roots,  and 
also  the  first  figure  or  figures  of  each.  We  may  then  approximate 
to  the  value  of  any  root  by  the  following 

RULE.  I.  Transform  the  given  equation  into  another  whose  roots 
shall  be  Itss  by  the  initial  figure  or  figures  of  the  required  root. 

II.  Divide  the  absolute  term  of  the   transformed  equation  by  the 
penultimate  <  o  efficient,  as   n  trial  iff  visor,  find   write  the  first  figure 
of  the  quotient  as  the  neoct  jigure  of  the  root  sought. 

III.  Transform  (he  last   equation   into  another  whose,  roots  shall 
be  less  than  those  of  the  previous  equation  by  the  figure  last  found  ; 
and  thus  continue  till  the  root   be  obtained  to  the  required  degree  of 
accuracy. 

NOTES.  1.  The  successive  transformations  required  in  obtaining  any 
root  may  all  be  made  in  a  single  operation  ;  and  for  the  sake  of  perspi- 
cuity, the  coefficients  obtained  in  each  transformation  may  be  marked  or 
numbered. 

2.  If  a  trial  figure  of  the  root,  obtained  by  any  division,  reduces  the 
absolute  term  X',  and  the  penultimate  coefficient  J£"'j,to  the  same  sign, 
this  figure  is  not  the  true  one,  and  must  be  diminished. 

3.  To  obtain  the  negative  roots,  it  will  be  most  convenient  to  change 
the  signs  of  the  alternate  terms  of  the   given  equation,  and  find  the. 
positive  roots  of  the    result;  these,  with    their   signs  changed,  will  be 
the  negative  roots  required. 

4.  —  If  the  penultimate  coefficient,  T',  should  reduce  to  zero  in  the  ope- 
ration, the  next  figure  of  the  root  may  be  obtained  by  dividing  the  ubanlnte 
term,  X',  by  the  coefficient  which  precedes  T',  and  extracting  the  square  root 
tf  the  quotient.    For,  if  T'  vanishes,  we  have,  in  the  transformed  equation, 


=  0;  or  y  =       —    , 
B* 


NUMERICAL    EQUATIONS   OF    HIGHER    DEGREES. 
EXAMPLES. 

1.  Given  x'— 2.x3— 2Qx— 40,  to  find  the  approximate  value  of  x. 

By  Sturm's  Theorem  we  find  that  this  equation  has  only  one 
real  root,  the  initial  figure  being  6.  We  now  obtain  the  decimal 
part,  to  2  places,  as  follows : 

OPERATION. 

—20  —4016.23 

24  24  ~ 


-f4  <»>  — 1G 

GO  13.448 


10  <»>  64  <*>  —2.552 

G  3.24 

67.24 

3.23 

1G.2  «>  70.52 


16.4 
.2 

«>   16.6 

EXPLANATION. — We  first  transform  the  given  equation  into  an- 
other whose  roots  are  less  i  y  6,  using  the  method  of  Synthetic  Di- 
vision, explained  in  (44:3).  The  coefficients  of  the  transformed 
equation  are  16,  64  and  — 16,  marked  (1)  in  the  operation.  Divid- 
ing the  absolute  term  — 16,  taken  with  the  contrary  sign,  by  the 
penultimate  coefficient  64,  we  obtain  .2,  the  next  figure  of  the  root. 

We  next  transform  the  equation  whose  coefficients  are  marked  (1), 
into  another  whose  roots  are  less  by  .2,  the  resulting  coefficients  be- 
ing marked  (2).  Dividing  2.552  by  70.52,  we  obtain  .03,  the  next 
figure  of  the  root.  The  operation  may  thus  be  continued  till  the 
root  is  obtained  to  any  required  degree  of  accuracy. 

2.  Given  x'+x* — 30x2 — 20x — 20  =  0,  to  find  one  value  of  x. 

By  Sturm's  Theorem    we  find  the  initial  figures  of  the  two  real 


HORNEK'S  METHOD.  419 

roots  to  be  5  and  — 5.  Changing  the  signs  of  the  alternate  terms 
of  the  equation,  we  obtain  the  decimal  part  of  the  negative  root,  by 
the  following 


OPERATION. 

IV. 

III. 

II. 

I. 

1  -1 

5 

—30 

20 

-f-  20 
—  5p 

—  20|5.731574 

—150 

-f-4 

—10 

—  30 

<i)_170.0000 

5 

45 

175 

(1)-fl45.000 

150.7071 

+35 

<2>—  10.2929 

5 

"""  \ 

83.153 

9.7727 

14 

(1)  105.00 

228.153 

<»>—  .5202" 

5 

13.79 

93.149 

.3304 

<"  19.0 

118.79 

<*>  321.302 

<4>—  .1898 

0.7 

14.28 

4.455 

.1653 

19.7 

133.07 

325.757 

<5>—  245 

.  i 

14.77 
«>  147.84 

4.474 

232 

20.4 

(3)  330.23 

.7 

.65 

.15 

13 

21.1 

148.49 

330.38 

0-      7 

.7 

.65 

.15 

<2>  21.8 

149.14 

<«>  330.5 

.65 

.1 

«>  150.  * 

330.6 

.1 

<4>  2  <«>  331. 

(6)  38  Am.  —5.731574. 

EXPLANATION. — "We  proceed  as  in  the  preceding  example  till  we 
obtain  the  terras  marked  (2),  in  the  operation.  Dividing  10.2929  by 
321.302,  we  obtain  .03  for  the  next  figure  of  the  root. 

At  this  point  we  commence  to  apply  decimal  contractions,  accord 
ing  to  the  principles  employed  in  the  contracted  method  of  cube 
root; -(24:3).  Let  it  be  observed,  that  ear Ji  contracted  term  in 
the  operation  contains  one  redundant  figure  at  the  right. 


420         NUMERICAL    EQUATIONS    OF   HIGHER    DEGREES. 


Commencing  with  column  IV,  we  have  21. 8  X -03  =  .65,  which 
added  to  column  III  gives  148.49.  Then  148.49X-03  =  4.455, 
which  added  to  column  II  gives  325.757.  Then  325.757  X  -03  = 
9.7727,  which  added  to  column  I  gives  — .5202.  Again  adding  .65 
to  column  III,  we  have  149.14.  Then  149.14X-03  =  4.474,  which 
added  to  column  II  gives  330.23,  after  dropping  one  place.  Again, 
adding  .65  to  column  III  gives  150  after  dropping  two  places.  In 
like  manner  we  continue  till  the  work  is  finished. 

NOTE. — Observe,  as  a  general  rule,  to  contract  the  several  columns,  for 
each  root  figure,  as  follows :— Column  I,  0  place ;  column  II,  1  place ; 
column  III,  2  places ;  column  IV,  3  places ;  and  so  on. 

Find  the  real  roots  of  the  following  equations : 

Ans.  5.1345787253. 


3...T8+2x2—  23x—  70  =  0. 
4.  »•—  x'+70x—  300  =  0. 


5. 


6. 


a—  500  =  0. 


=  0. 


7.  rc«—  4z9—  24x+  48  =  0. 


8.  z4-f-z'4-za—  x—  500  =  0. 


Ans. 


Ans 


Ans.  3.7387936782. 
Ans.  7.6172797559. 

(      3.3792053825, 
4.5875359541, 
—6.9667413367. 

(      1.7191292611, 

.   <      6.5461457261. 

1—4.2652749871. 


Ans. 


(      4.4604168201, 
}  —4.9296646474. 


9.  x4—  9*'—  llza— 


10.  a4— 


=  0.       Ans 


.  «f 


=  0. 


Ans. 


.1796840250, 
10.2586086356. 

2.8580833082, 

.6060183069, 

.4432769396, 

—3.9073785547. 


11.  x«__ 


=  0. 


Ans.   < 


/-—3.0653157918, 
\_  .6915762805, 
-  .1756747993. 

)        .8795087084, 
\     3.0530581627, 


NOTE. — Full  solutions  of  the  examples  above  may  be  found  in  the  Key. 


•p 

; 

' 


>>\v> 


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This  book  is  DUE  on  the  last  date  stamped  below. 


OCT  21   1947 

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esit 


